Deutsch just referred me to this paper which is the more formal formulation of his multiple-worlds theory. I must confess that on a cursory read it seems to be a compelling argument. So I may have to rethink this whole thing.

That paper is entitled "The Structure of the Multiverse" and its abstract is delightfully succinct. I quote it here in its entirety:

The structure of the multiverse is determined by information flow.

Those of you who have been following my quantum adventures know that I am a big fan of information theory, so I was well primed to resonate with Deutsch's theory. And I did resonate with it (and still do). Deutsch's argument was compelling (and still is). Nonetheless, I never wrote a followup for two reasons. First, something was still bothering me about the argument, though I couldn't really put my finger on it. Yes, Deutsch's argument was compelling, but on the other hand, so was my argument (at least to me). The difference seemed to me (as many things in QM interpretations do) a matter of taste, so it seemed pointless to elaborate. And second, I didn't think anyone reading this blog would really care. So I tabled it.

But last May the comment thread in the original post was awakened from its slumber by a fellow named Elliot Temple. The subsequent exchange led me to this paper, of which I was previously unaware. Here's the abstract, again, in its entirety:

The probabilistic predictions of quantum theory are conventionally obtained from a special probabilistic axiom. But that is unnecessary because all the practical consequences of such predictions follow from the remaining, non-probabilistic, axioms of quantum theory, together with the non-probabilistic part of classical decision theory.

The "special probabilistic axiom" to which Deutsch refers is called the Born rule (named after Max Born). The "remaining, non-probabilistic axioms of quantum theory" comprises mainly the Schrödinger equation. (To condense things a bit I'll occsaionally refer to these as the BR and the SE.)

The process of applying quantum mechanics to real-world situations consists of two steps: first you solve the SE. The result is something called a "wave function". Then you apply the BR to the wave function and what pops out is a set of probabilities for various possible results of the experiment you're doing. Following this procedure yields astonishingly accurate results: no experiment has ever been done whose outcome is at odds with its predictions. The details don't matter. What matters is: there's this procedure. It yields incredibly accurate predictions. It consists of two parts. One part is deterministic, the other part isn't.

This naturally raises the question of why this procedure works as well as it does. In particular, why does the procedure have two parts? And why does it only yield probabilities? Answering these questions is the business of "interpretations" of quantum mechanics. Wikipedia lists almost twenty of these. The fact that after nearly 100 years no consensus has emerged as to which one is correct gives you some idea of the thorniness of this problem.

So the paper that Elliot referred me to was potentially a Big Deal. It is hard to overstate the magnitude of the breakthrough this would be. It would show that there are not in fact two disparate parts to the theory, there is only one: the SE. Such a unification would be of the same order of magnitude as the discovery of relativity. It would be headline news. David Deutsch would be a Nobel Laureate, on a par with Newton and Einstein. But the fact that there is still an active debate over the issue shows that Deutsch's claim has not been universally accepted. So there would seem to be only two possibilities: either Deutsch is wrong, or he's right and the rest of the physics community has failed to recognize it.

Normally when a claim of a major result like this fails to be recognized by the community it's because the claim is wrong. In fact, more than 99% of the time it's because the claimant is a crackpot. But Deutsch is no crackpot. He's a foundational figure in quantum computing. He discovered the first quantum algorithm. Even if he got something wrong he very likely got it wrong in a very interesting way.

So I decided to do a deep dive into this. It led me down quite the little rabbit hole. There are a number of publishedcritiques of Deutsch's work, and counter-critiques critiquing the critiques, and counter-counter-critiques. They're all quite technical. It took me a couple of months of steady effort to sort it all out, and that only with the kind of help of a couple of people who understand all this stuff much better than I do. (Many thanks to Tim Maudlin, David Wallace, and especially the patient, knowledgeable, and splendidly-pseudonymed /u/ididnoteatyourcat on Reddit.)

In the rest of this post I'm going to try to describe the result of going down that rabbit hole in a way that is accessible to what I think is the majority of the audience of this blog. The TL;DR is that Deutsch's argument depends on at least one assumption that is open to legitimate doubt. Figuring out what that assumption is isn't easy, and whether or not the assumption is actually untrue is arguable. That's the reason that Deutsch hasn't won his Nobel yet.

I have to start with a review of the rhetoric of the many-worlds interpretation of quantum mechanics (MWI). The rhetoric says that when you do a quantum measurement it is simply not the case that it has a single outcome. Instead, what happens is that the universe "splits" into multiple parts when a measurement is performed, and so all of the possible outcomes of an experiment actually happen as a matter of physical fact. The reason you only perceive a single outcome is that you yourself split into multiple copies. Each copy of you perceives a single outcome, but the sum total of all the "you's" that have been created collectively perceive all the possible outcomes.

I used the word "rhetoric" above because, as we shall see, there is a disconnect between what I have just written and the math. To be fair to Deutsch, his rhetoric is different from what I have written above, and it more closely matches the math. Instead of "splitting", on Deutsch's view the universe "peels apart" (that's my terminology) in "waves of differentiation" (that is Deutsch's terminology) rather than "splitting" (that is everyone else's terminology) but this is a detail. The point is that at the end of a process that involves you doing a quantum measurement with N possible outcomes, there are, again in point of actual physical fact, N "copies" of you (Deutsch uses the word "doppelgänger").

Again, to be fair to Deutsch, he acknowledges that this is not quite correct:

Universes, histories, particles and their instances are not referred to by quantum theory at all – any more than are planets, and human beings and their lives and loves. Those are all approximate, emergent phenomena in the multiverse. [The Beginning of Infinity, p292, emphasis added.]

All of the difficulty, it will turn out, hinges on the fidelity of the approximation. But let us ignore this for now and look at Deutsch's argument.

Deutsch attempts to capture the idea of probability in a deterministic theory using game theory, that is, by looking at how a rational agent should act, applying a few reasonable-looking assumptions about the utility function, and showing that a rational agent operating under the MWI would act exactly as if they were using the Born rule. The argument is long and technical, but it can be summarized very simply.

[Note to nit-pickers: this simplified argument is in fact a straw man because it is based on the assumption that branch counting is a legitimate rational strategy, which is actually false on the Deutsch-Wallace view. But since the conclusion I am going to reach is the same as Deutsch's I consider this legitimate rhetorical and literary license because the target audience here is mainly non-technical.]

For simplicity, let's consider only the case of doing an experiment with two possible outcomes (let's call them A and B). The game-theoretical setup is this: you are going to place a bet on either A or B and then do the experiment. If the outcome matches your choice, you win $1, otherwise you lose $1.

If the experiment is set up in such a way that the quantum-mechanical odds of each outcome are the same (i.e. 50-50) then there is no conflict between the orthodox Born-rule-based approach and the MWI: in both cases, the agent has no reason to prefer betting on one outcome over the other. The only difference is the rationale that each agent would offer: one would say, "The Born rule says the odds are even so I don't care which I choose" and the other would say, "I am going to split into two and one of me is going to experience one outcome (and win $1) and the other of me is going to experience the other outcome (and lose $1), and that will be situation no matter whether I choose A or B, so I don't care which I choose."

[Aside: Deutsch goes through a great deal more complicated argument to prove this result because it is based on an assumption that Deutsch rejects. In fact, he goes on from there to put in a great deal more effort to extend this result to an experiment with N possible outcomes, all of which have equal probabilities under the Born rule. He has to do this because my argument is based on a tacit assumption that Deutsch rejects. We'll get to that. My goal at this point is not to reproduce Deutsch's reasoning, only to convince you that this intermediate result is plausibly true.]

Now consider a case where the odds are not even. Let's arrange for the probabilities to be 2:1 in favor of A (i.e. A happens 2/3 of the time, B happens 1/3 of the time, according to the Born rule). Now we have a disconnect between the two world-views. The Bornian would obviously choose A. But what possible reason could the many-worlder have for doing the same? After all, the situation is unchanged from before: again the many-worlder is going to split into two (because there are still only two possible outcomes). What possible basis could they have for preferring one outcome over the other that doesn't assume the Born rule and hence beg the question?

Deutsch's argument is based on an assumption called branching indifference. Deutsch himself did not make this explicit in his original paper, it was clarified by David Wallace in a follow-up paper. Branching indifference says that a rational agent doesn't care about branching per se. In other words, if an agent does a quantum experiment that doesn't have a wager associated with it, then the agent has no reason to care whether or not the experiment is performed or not.

The reasoning then proceeds as follows: suppose that the many-worlder who ends up on the A branch does a follow-up experiment with two outcomes and even odds, but without placing a bet. Now there are three copies of him, two of which have won $1 and one of which has lost $1. But (and this is the crucial point) all of these copies are now on branches that have equal probabilities. Because of branch indifference, this situation is effectively equivalent to one where there was a single experiment with three outcomes, each with equal probability, but two of which result in winning $1, and where the agent had the opportunity to place the bet on both winning branches.

So that sounds like a reasonable argument. In fact, it is a correct argument, i.e. the conclusions really do follow from the premises.

But are the premises reasonable? Well, many many-worlders think so. But I don't. In particular, I cast a very jaundiced eye on branching indifference. There are two reasons for this. But first, let's look at Wallace's argument for why branching indifference is reasonable:

Solution continuity and branching indifference — and indeed problem continuity — can be understood in the same way, in terms of the limitations of any physically realisable agent. Any discontinuous preference order would require an agent to make arbitrarily precise distinctions between different acts, something which is not physically possible. Any preference order which could not be extended to allow for arbitrarily small changes in the acts being considered would have the same requirement. And a preference order which is not indifferent to branching per se would in practice be impossible to act on: branching is uncontrollable and ever-present in an Everettian universe.

If that didn't make sense to you, don't worry, I'll explain it. But first I want to take a brief diversion. Trust me, I'll come back to this.

Remember how I said earlier that my simplified argument for Deutsch's conclusion was based on a premise that Deutsch would reject? That premise is called branch counting. It is the idea that the number of copies of me that exist matters. This seems like an odd premise to dispute. How could it possibly not matter if there is one of me winning $1 or a million of me each winning $1? The latter situation might not have a utility that is a million times higher than the former, but if I'm supposed to care about "copies of me" at all, how can it not matter how many there are?

Here is Wallace's answer:

Why it is irrational: The first thing to note about branch counting is that it can’t actually be motivated or even defined given the structure of quantum mechanics. There is no such thing as “branch count”: as I noted earlier, the branching structure emergent from unitary quantum mechanics does not provide us with a well-defined notion of how many branches there are.

Wait, what??? There is no "well defined notion of how many branches there are?"

No, there isn't. Wallace reiterates this over and over:

...the precise fineness of the grain of the decomposition is underspecified

There is no “real” branching structure beyond a certain fineness of grain...

...agents branch all the time (trillions of times per second at least, though really any count is arbitrary)

...in the actual physics there is no such thing as a well-defined branch number

Remember how earlier I told you that there was a disconnect between the rhetoric and the math? That the idea of "splitting" or "peeling apart" or whatever you want to call it was an approximation? Well, this is where the rubber meets the road on that approximation. Branching indifference is necessary because branching is not a well-defined concept.

So what about the rhetoric of MWI, that when you do an experiment with N possible outcomes that you split/peel-apart/whatever-you-want-to-call-it into N copies of yourself? That is an approximation to the truth, but like classical reality itself, it is not the truth. The actual truth is much more complex and subtle, and it hinges on what the word "you" means.

If by "you" you mean your body, which is to say, all the atoms that make up your arms and legs and eyes and brain etc. then it's true that there is no such thing as a well-defined branch count. This is because every atom — indeed, every electron and every other sub-atomic particle — in your body is constantly "splitting" by virtue of its interactions with other nearby particles, including photons that are emitted by the sun and your smart phone and all the other objects that surround you. These "splits" propagate out at the speed of light and create what Deutsch calls "waves of differentiation", what I call the "peeling apart" of different "worlds". (If you are a regular reader you will have heard me refer to this phenomenon as creating "large systems of mutually entangled particles". Same thing.) This process is a continuous one. There is never a well-defined "point in time" where the entire universe splits into two, and no point in time where you (meaning your body) splits into two. There is a constant and continuous process of "peeling apart". Actually many, many (many!) peelings-apart, all of which are happening continuously. To call it mind-boggling would be quite the understatement.

On the other hand, if by "you" you mean "the entity that has subjective experiences and makes decisions based on those experiences" then things are much less clear. I don't know about you, but my subjective experience is that there is exactly one of me at all times. I consider this aspect of my subjective experience to be an essential component of what it means to be me. I might even go so far as to say that my subjective experience of being a single unified whole defines what it is to be "me". So the only way that there could be a "copy of me" is if there is another entity that has a subjective experience that is bound to the same past as my own, but whose present subjective experience is somehow different from my own e.g. my experiment came out A and theirs came out B. An entity whose subjective experience is indistinguishable from my own isn't a copy of me, it's me.

The mathematical account of universes "peeling apart" has nothing to say about when the peeling process has progressed far enough to be considered a fully-fledged universe in its own right and so it has nothing to say about when I have "peeled apart" sufficiently to be considered a copy. That is why branch count is not a coherent concept.

And yet, if I am going to apply the notion of branching to myself (which is to say, to the entity having the subjective experience of being a coherent and unified whole) then branch count must be a coherent concept. It might not be possible to know the branch count, but at any point in time whatever underlying physical processes are really going on, it has to either qualify as me branching or not. There is no middle ground.

So we are faced with this stark choice: we can either believe the math, or we can believe our subjective experiences, but we can't do both, at least not at the same time. We can take a "God's eye view" and look at the universal wave function, or we can take a "mortal's-eye view" and see our unified subjective experience as real. But we can't do both simultaneously. It's like a Necker cube. You can see it one way or the other, but not both at the same time.

Interestingly, this is all predicted by the math! In fact, the math tells us why there is this dichotomy. Subjective experience is necessarily classical because it requires copying information. In order to be conscious, you have to be conscious of something. In order to make decisions, you have to obtain information about your environment and take actions that affect your environment. All of these things require copying information into and out of your brain. But quantum information cannot be copied. Only classical information can be copied. And the only way to create copyable classical information out of a quantum system is to ignore part of the quantum system. Classical behavior emerges from quantum systems (mathematically) when you trace over parts of the system. Specifically, it emerges when you consider a subset of an entangled system in isolation from the rest of the system. When you do that, the mathematical description of the system switches from being a pure state to being a mixed state. Nothing physical has changed. It's purely a question of the point of view you choose to take. You can either look at the whole system (in which case you see quantum behavior) or you can look at part of the system (in which case you see classical behavior) but you can't do both at the same time.

As a practical matter, in our day-to-day lives we have no choice but to "look" only at "part" of the system, because "the system" is the entire universe. (In fact, it's an interesting puzzle how we can observe quantum behavior at all. Every photon has to be emitted by, and hence be entangled with, something. So why does the two-slit experiment work?) We can take a "God's-eye view" only in the abstract. We can never actually know the true state of the universe. And, in fact, neither can God.

Classical reality is what you get when you slice-and-dice the wave function in a particular way. It turns out that there is more than one way to do the slicing-and-dicing, and so if you take a God's-eye view you get more than one classical universe. An arbitrary number, in fact, because the slicing-and-dicing is somewhat arbitrary. (It is only "somewhat" arbitrary because there are only certain ways to do the slicing-and-dicing that yield coherent classical universes. But even with that constraint there are an infinite number of possibilities, hence "no well-defined branch count".) But the only way you can be you, the only way to become aware of your own existence, indeed the only way to become aware of anything, is to descend from Olympus, ignore parts of the wave function, and become classical. That leaves open the question of which parts to ignore. To me, the answer is obvious: I ignore all of it except the parts that measurably effect the "branch" that "I" am on. To me, that is the only possible rational choice.

89 comments:

> Now consider a case where the odds are not even. Let's arrange for the probabilities to be 2:1 in favor of A (i.e. A happens 2/3 of the time, B happens 1/3 of the time, according to the Born rule). Now we have a disconnect between the two world-views. The Bornian would obviously choose A. But what possible reason could the many-worlder have for doing the same? After all, the situation is unchanged from before: again the many-worlder is going to split into two (because there are still only two possible outcomes).

The reason is simple. Not all universes exist in equal quantities. There could be e.g. twice as many of universe A as of universe B. So he’d prefer to make the bet where he wins in 2/3 of (the relevant branch of) the multiverse over the one that wins in 1/3 of the multiverse. More copies of him will win.

The idea of there being exactly one copy of a person for each type of universe (e.g. win bet type or lose bet type) is incorrect.

There are many identical universes. When they “split” (aka “branch” or “differentiate”), the split does not have to happen with equal proportions.

You don’t split from one into two. You split from many into many (the same number as before – no universes are created nor destroyed). “Splitting” means some universes become different that were, previously, identical. You can split into e.g. 2/3 of one outcome and 1/3 another outcome.

(Warning: speaking about universes is an approximation. They have no fundamental role in physics. One of the reasons is along the lines Ron states in the quote below that begins “This process is a continuous one.” Plus universes are big but quantum physics is local – change spreads at the speed of light or less.)

Wallace writes:

https://arxiv.org/pdf/0906.2718.pdf

> **Branching Indifference:** An agent doesn’t care about branching per se: if a certain measurement leaves his future selves in N different macrostates but doesn’t change any of their rewards, he is indifferent as to whether or not the measurement is performed.

Ron writes:

> Branching indifference says that a rational agent doesn't care about branching per se. In other words, if an agent does a quantum experiment that doesn't have a wager associated with it, then the agent has no reason to care whether or not the experiment is performed or not.

This is unclear or is different than Wallace because it speaks of a *wager* rather than a *reward*. A wager means betting money, whereas a reward is anything that is good or bad according to an agent’s preferences.

Wallace is basically saying that an agent is indifferent to branching into copies *if*, after branching, no copy of the agent is worse (or better) off in any way.

That sounds totally unobjectionable to me. If a particular branching has no downsides or upsides (in any universe), according to an agent’s preferences, then an agent is indifferent to it. In other words, agents care about positive and negative rewards rather than branching itself (branching sometimes, but not always, has consequences for rewards).

Being pedantic, one could say the same thing about any other physical events (eating breakfast, getting fired, hitting a homerun, winning the lotto, etc.) – it’s not the event itself, per se, which matters, but whether or not there is any change in reward according to the agent’s preferences. E.g. “sleep indifference” states that agents are indifferent to how much sleep they get, per se, as long as there is no positive or negative change to their rewards.

In other words, if an agent has no preference about something, then that agent is indifferent to it. And agents don’t necessarily have preferences about all possible multiversal branching.

> if I'm supposed to care about "copies of me" at all, how can it not matter how many there are?

You can care about a *measure* of copies that isn’t simple quantity/count (a quantity/count of marbles is an example of simple quantity/count).

> So what about the rhetoric of MWI, that when you do an experiment with N possible outcomes that you split/peel-apart/whatever-you-want-to-call-it into N copies of yourself?

That is *not* the MWI position. You split into N *different versions* of yourself, but not into N copies of yourself (that’s different because it could be e.g. 3 copies of the first version, 8 copies of the second version, etc. But they’re harder to measure than simple counting.)

> This process is a continuous one. There is never a well-defined "point in time" where the entire universe splits into two, and no point in time where you (meaning your body) splits into two.

Agreed.

> subjective

I think it’s better to start by considering how physics works without worrying about conscious or “subjective” experiences. Let’s discuss all the other stuff first and only tackle issues related to consciousness after agreeing on the rest (like what happens with dice, photons, mirrors and cups of water). That’s what I’ve done above.

> You can either look at the whole system (in which case you see quantum behavior) or you can look at part of the system (in which case you see classical behavior) but you can't do both at the same time.

But if you look in certain parts you do *not* see classical behavior. Classical physics being *false* and *refuted by some experiments* is why we have quantum theory. Classical physics is a good approximation in many cases people encounter in their daily lives, but not in all cases. Agreed?

I'm going to respond out of order, saving the most important point for last.

> You split into N *different versions* of yourself, but not into N copies of yourself

I don't normally like quibbling over terminology, but in this case I agree that "versions" is a better, more descriptive term than "copies" here.

> You don’t split from one into two. You split from many into many

Yes, I understand that that's what the math says. The problem is that I don't feel like many. I feel like one. If I am many, why don't I feel like it?

Also, if I'm many, how many am I?

(Feel free to consider those rhetorical questions. We don't actually disagree on the physics, we disagree on the philosophy. See below.)

> Not all universes exist in equal quantities.

Yes, but what do those quantities have to do with *decisions*? You can't just *assume* that "higher-quantity" universes should have a greater weight in decision-making. That is begging the question.

> This is unclear or is different than Wallace because it speaks of a *wager* rather than a *reward*.

A wager is a specific kind of decision. Deutsch frames his argument in terms of decision theory. So speaking of wagers should not be too surprising.

But I'm happy to just substitute "no reward" for "no wager" here. It amounts to the same thing in this case.

> it’s not the event itself, per se, which matters, but whether or not there is any change in reward according to the agent’s preferences

Yes, I agree with that. The problem is that in a branching universe the phrase "the agent" no longer has a well-defined referent.

> Classical physics is a good approximation in many cases people encounter in their daily lives, but not in all cases. Agreed?

Yes, of course.

> I think it’s better to start by considering how physics works without worrying about conscious or “subjective” experiences.

This is our biggest disconnect. My subjective experience is the only data I have direct access to. It's the only reason I have to even suspect that there is such a thing as "the laws of physics" out there to be discovered. So you can start by considering how physics works if you like, but if you want to tell a complete story of how the world works then sooner or later you're going to have to circle back and consider how *considering* works.

>> I think it’s better to start by considering how physics works without worrying about conscious or “subjective” experiences.>> This is our biggest disconnect. My subjective experience is the only data I have direct access to. It's the only reason I have to even suspect that there is such a thing as "the laws of physics" out there to be discovered. So you can start by considering how physics works if you like, but if you want to tell a complete story of how the world works then sooner or later you're going to have to circle back and consider how *considering* works.

Considering is a form of information processing. A universe is a structure within the multiverse where information can flow from one system to another. For example, there are versions of me sitting 1 inch to the right of my current position. I can't exchange information with those other version of me. I can't see whether one of those versions of me is sitting with his legs crossed. That is why those versions of me count as separate versions in separate universes.

Now, I have a record in my memory of writing the first version of this paragraph. There is also information in the environment about what I typed, e.g. - information in sound waves from my typing in light reflecting off the keys and so on. This information can be used to decide whether I am the same person as the person who wrote the first paragraph of this reply. Other versions of me wrote a different second paragraph but also have records of the same first paragraph. The fact that there is more than one version of me with such a record doesn't change the fact that the identification can be made.

A similar story about identifying objects using records can be told about other objects like the keyboard I'm typing on, the pen sitting on the desk behind the keyboard and so on. Any decision you're going to make has to use the same kinds of records that would be used to identify the pen or the keyboard or whatever. From the point of view of physics, there is no particular reason to make a special case for people as opposed to pens or computers or whatever. So we might as well consider a computer programmed to maximise its rewards rather than a person. We know how to program a computer to just follow a particular rule. Programming a person to just follow a particular rule is difficult and raises irrelevant moral problems, so discussing a computer program makes more sense.

> So is there any reason I should *care* about the existence other versions of me (except perhaps as an intellectual curiosity)?

The only direct reason to be interested in the existence of other versions of you is wanting to have a consistent worldview that actually explains how the world works. If you don't care about this you will make many more uncorrected mistakes since you remove constraints on your ideas that might improve them. This will include mistakes in actions you take directly and mistakes in picking politicians, advisors etc. To understand this issue read the title essay from Ayn Rand's book "Philosophy: Who Needs It".

> So is there any reason I should *care* about the existence other versions of me (except perhaps as an intellectual curiosity)?

The "you" that places the wager consists of multiple identical copies. For simplicity, we'll call it 100. So there are 100 clones of you which, as a group, place a wager. Why would "you" care about "other versions" of you? Because "you" (as of the start of the scenario where you make a choice about a wager) are 100 people, you should bet in a way that gives the best outcome for those 100 people.

You should try to maximize outcomes for the versions of you that placed the wager, precisely because they are part of the entity that placed the wager. Each of them is a real person who is making a wager and wants to win. If you bet so that 2/3 of them win, that that's better for the group, and also then each of them individually has a 2/3 chance to win rather than e.g. a 1/3 chance.

PS Because your captcha system seems to block some of my comments and also keeps presenting me with many captchas in a row (e.g. 5 in a row, I've done ~two dozen in total), I don't want to continue discussing here. If you want to discuss further, please post at https://curi.us/2209-physics-discussion where you can post without any captcha or moderation, and with better formatting too. If you don't care enough to do that then I think I'll, sadly, give up because the software here is too broken and/or user hostile.

Alan: The only direct reason to be interested in the existence of other versions of you is wanting to have a consistent worldview that actually explains how the world works. [Emphasis added.]

elliot: Why would "you" care about "other versions" of you? Because "you" (as of the start of the scenario where you make a choice about a wager) are 100 people, you should bet in a way that gives the best outcome for those 100 people.

So which is it?

> wanting to have a consistent worldview

There is nothing inconsistent in Bohmian mechanics or GHZ collapse, not even, for that matter, in the Copenhagen interpretation. So this can't be the reason.

> The "you" that places the wager consists of multiple identical copies. For simplicity, we'll call it 100.

Call it whatever you like, this cannot be a correct explanation. If I am N identical copies, then after O(log(N)) splits I will be 1. What happens then?

Note also that, according to Wallace, splits happen "trillions of times per second at least". (That's an actual quote, not a scare quote.) I am approximately 10^9 seconds old, so I have already undergone at least 10^20 splits, so when I started out, I must have consisted of at least 2^10^20 copies. And in fact it is even worse than that because most splits are not the result of doing well-controlled spin measurements using a Stern-Gehrlach apparatus, it's the result of natural decoherence, which cause N-way splits for very large values of N. So "call it 100" is a laughably inaccurate oversimplification. And the implication that the number of copies that you consist of at any one time is a finite integer is simply false because, again as Wallace points out, branch count is not well-defined.

From the article you linked to:

> The short, approximate version is that it's better to win $1 in 66 universes and lose $1 in 34 than vice versa.

This is branch-counting, and Wallace himself debunks this in his paper. If it were valid, then branching-indifference would not hold and Deutsch's proof would not carry through. So this is not a "short, approximate version", it is simply wrong.

> your captcha system seems to block some of my comments and also keeps presenting me with many captchas in a row

It's not my captcha system, it's Google's. But it has a bug that allows it to be easily defeated if you're posting from a blogger account (which you are): when a captcha appears, just ignore it. Click anywhere outside the captcha. It will disappear. The checkbox next to "I'm not a robot" will not be checked, but the publish button should work at this point nonetheless. At least that works for me.

FWIW, I just checked my settings and I actually have the captcha disabled, so the fact that it appears at all is a bug in Blogger (which might account for why it is so easily defeated). I'm sorry about that, but this is something over which I obviously have no control. Blogger's comment system has always been annoying in one way or another, and I've considered switching to another platform on a number of occasions, but I've been on blogger for over fifteen years now (since long before Google bought them) so the inertia is high.

(this is the linked comment in 2 parts. i tried again with no captcha and discovered it was slightly longer than the length limit. i think that is why it wouldn't post before, except that i did not get the error message when i was using the captcha.)

> Yes, I understand that that's what the math says. The problem is that I don't feel like many. I feel like one. If I am many, why don't I feel like it?

MWI is an *objective* theory based on *scientific observations and math*, **not** based on a philosophical theory of consciousness. While *I* could comment on the matter, and DD also has opinions on the matter, they are a separate issue than MWI.

If you told me that due to misunderstandings of consciousness (which you could name and explain), some of experimental data needed to be reconsidered or rejected, that would be relevant. Same if you rejected some math, logic or prior-to-QM physics claims. But if you accept all the experimental data and all the math, logic and physics premises, then doesn't MWI (or a few other options which mostly differ from MWI by their claims about objective reality, not by their theories of consciousness) follow?

> Yes, but what do those quantities have to do with *decisions*? You can't just *assume* that "higher-quantity" universes should have a greater weight in decision-making. That is begging the question.

The point of DD's paper is to prove that, in terms of betting by the "rational agents" of decision theory, you should do betting-decision-making according to "higher-quantity" universes (in order to maximize multiverse-wide betting returns). He does not assume this point nor beg this particular question. Your objections related to consciousness do not constitute a criticism of DD's proof of this matter from his premises.

The short, approximate version is that it's better to win $1 in 66 universes and lose $1 in 34 than vice versa. What they have to do with decisions is that you (a group of many fungible, identical, indistinguishable instances of a person) would prefer that more of you (more of those instances that already exist and are part of the group of instances that make the decision) would prefer that more of your instances win over fewer winning.

> The problem is that in a branching universe the phrase "the agent" no longer has a well-defined referent.

Do you mean because an agent has microscopic changes which haven't propagated to his entire body at once (before some changes finish propogating, others begin, so there are always some changes which have not propgated to his entire body)? (And the same point works with just his brain.) Is that the issue?

I think that issue of defining a particular agent and its preferences in a world that is in constant flux is not really a QM or MWI issue. Heraclitus could have made a similar complaint. And I don't think other QM interpretations, which do not contradict the math and observations, will change this problem much. DD's point in the paper is that *if* you accept certain claims of decision theory (which are pretty widely accepted), including that you can take as a starting point an agent with a well-defined set of preferences, *then* various things follow. The details of how to deal with a world (including agents) in constant flux is a separate matter not covered in that paper.

I think this separate matter of dealing with flux is widely believed to be soluble, and that DD and I have that belief. And I think it can be approached without getting into the problems of consciousness or subejctivism that I avoided above. Like you could have a software agent, which isn't even (general) intelligent, and it bets trying to maximize some function, and it too would have the constantly-undergoing-change issue. The solution, in short, is that despite the ongoing flux/chaos/change *in some respects*, there are some things which change very very little over some short time periods, so they can be taken as approximately constant at that time when they are approximately unchanging.

Something being approximately constant, rather than exactly constant, means that error correction is needed, which is getting pretty far afield (though covered a bit in BoI which explains the advantage of digital over analog for error correction). Very briefly it's like how computer circuits deal with electrical signals that are approximate, not exact, in terms of strength and timing. Minor fluctuations can be and are dealt with.

Feel free to put content somewhere else and just post links here if that works better for you.

> I'll keep trying here.

Happy to hear that.

> if you accept all the experimental data and all the math, logic and physics premises,

I do, but note that I include my own subjective experience in the "experimental data". In fact, as I've pointed out before, my subjective experience is in fact the *only* experimental data I have direct access to. That includes the subjective experience of interacting with other people, reading physics papers. And yes, I know about DD's argument that indirect evidence is *better* than direct evidence, and I agree with that argument. Nonetheless, my subjective experience is still something that a complete theory of reality needs to account for as far as I'm concerned.

> then doesn't MWI (or a few other options which mostly differ from MWI by their claims about objective reality, not by their theories of consciousness) follow?

There are at least four interpretations of QM that are widely considered viable in the sense that the cannot be ruled out on the evidence alone: MWI, Bohm, GRW collapse, and Copenhagen. I personally subscribe to a fifth interpretation which is similar enough to MWI that I'm pretty confident that it is logically viable even though it is not as popular as the other four, but that's neither here nor there. If the case for MWI were the kind of slam-dunk that you seem to feel that it is, there would not be any controversy. It would be widely considered to be settled science, as uncontroversial as relativity. But it isn't.

One cannot definitively rule out the possibility that MWI is in fact the only logically tenable interpretation and that the majority of the physics community is just too stupid to understand this. My Bayesian prior on this is pretty low.

> The point of DD's paper is to prove that, in terms of betting by the "rational agents" of decision theory, you should do betting-decision-making according to "higher-quantity" universes (in order to maximize multiverse-wide betting returns). He does not assume this point nor beg this particular question.

No, that's not true. What DD's paper shows is that *if* a rational agent accepts the quality metric of maximizing multiverse-wide betting returns biased according to branch *weights* (and not branch *counts*) *then* a rational agent should behave *as if* the Born rule were true.

> Your objections related to consciousness do not constitute a criticism of DD's proof of this matter from his premises.

You're right, because my criticism is not a criticism of the claim that Deutsch actually makes (as I have re-stated it above).

Here, verbatim, is the claim Deutsch actually makes:

"[A]ll the practical consequences of [probabilistic quantum] predictions follow from the remaining, non-probabilistic, axioms of quantum theory, together with the non-probabilistic part of classical decision theory"

And I accept that, but with one important and necessary proviso: an agent has to adopt the correct utility function. It is in the definition of the utility function that Deutsch begs the question. Yes, if you adopt a utility function that is logically equivalent to the Born rule then it is no surprise that the resulting rational decisions will be the same as if the agent had simply adopted the Born rule. But Deutsch never justifies his utility function. Wallace tries to, but he fails. He has to assume branching indifference, and that, it turns out, is where the Born rule is hiding.

(This is no small thing, BTW. Reducing the Born rule to branching indifference is significant progress, and I think actually leads to some deep insights. But it doesn't actually solve the problem.)

> The short, approximate version is that it's better to win $1 in 66 universes and lose $1 in 34 than vice versa.

Yes, that's obviously true. The problem is that if these 100 universes already existed before the experiment was done, then that completely undermines the argument that led us to believe in quantum theory in the first place. Here's a quote from TFoR:

"Could it be that the photon splits into fragments which, after passing through the slits, change course and recombine? We can rule that possibility out too. If, again, we fire one photon through the apparatus, but use four detectors, one at each slit, then at most one of them ever registers anything. Since in such an experiment we never observe two of the detectors going off at once, we can tell that the entities that they detect are not splitting up."

The only reason there was a problem in the first place is because a photon was taken (on good evidence) to be an indivisible unit that could not "split up". So which is it? Can a photon split up or not?

The answer (as I explained in the OP) is that it depends on your point of view. From a God's-eye point of view, it can. From a mortals-eye point of view, it can't.

> > The problem is that in a branching universe the phrase "the agent" no longer has a well-defined referent.

> Do you mean because an agent has microscopic changes which haven't propagated to his entire body at once

No, I mean that "the agent" can refer to either the pre-split agent, or any one of the N post-split agents. If you're going to talk about *the* agent (singular) then you have to specify which of the N+1 agents you mean.

> Heraclitus could have made a similar complaint.

Indeed, but he wasn't trying to do physics. If you're going to do that, you should expect to be held to a higher standard.

> I think this separate matter of dealing with flux is widely believed to be soluble

And I'm pretty sure that you're wrong. In fact, I'm pretty sure that I can *prove* that you're wrong (or at least advance a pretty compelling argument)! It might even be a publishable result. But here's a preview:

> Minor fluctuations can be and are dealt with.

Dealing with "minor fluctuations" often requires a radical departure from the conventional wisdom. Explaining the precession of the orbit of Mercury requires a lot more than a minor tweak to Newtonian mechanics. And going from the discrete to the continuous can be highly problematic, particularly if you think about it in terms of information theory.

> Call it whatever you like, this cannot be a correct explanation. If I am N identical copies, then after O(log(N)) splits I will be 1. What happens then?

OK, you want less approximate. Some BoI quotes (not in order):

> Thus the information in the fictional multiverse flows along a branching tree, whose branches – histories – have different thicknesses (measures) and never rejoin once they have separated.

> *Instances* In parts of the multiverse that contain universes, each multiversal object consists approximately of ‘instances’, some identical, some not, one in each of the universes.

> In quantum physics, information flow in the multiverse is not as tame as in that branching tree of histories I have described. That is because of one further quantum phenomenon: under certain circumstances, the laws of motion allow histories to rejoin (becoming fungible again).

> In principle, a phenomenon could appear unpredictable to observers for one or more of three reasons. [...] The third – which had never been imagined before quantum theory – is that two or more initially fungible instances of the observer become different.

> Then they know that, when they run the transporter, an infinite number of fungible instances of themselves, all sharing the same history, are doing so at the same time.

> Our fictional theory has not provided enough structure in its multiverse to give a meaning to ‘half the universes’, but the real quantum theory does. As I explained in Chapter 8, the method that a theory provides for giving a meaning to proportions and averages for infinite sets is called a *measure*. A familiar example is that classical physics assigns *lengths* to infinite sets of points arranged in a line. Let us suppose that our theory provides a measure for universes.

You don't actually count universes, you measure them, just like you don't count points, you measure them.

You don't run out of points when you divide up an inch repeatedly. Same with instances in the multiverse.

> Nonetheless, my subjective experience is still something that a complete theory of reality needs to account for as far as I'm concerned.

Yes, but there is nothing about MWI which contradicts any of your subjective experiences. It never said "You will never subjectively experience X" for any X that you have experienced.

> No, that's not true. What DD's paper shows is that *if* a rational agent accepts the quality metric of maximizing multiverse-wide betting returns biased according to branch *weights* (and not branch *counts*) *then* a rational agent should behave *as if* the Born rule were true.

If a multiversal agent in N states tries to maximize the outcomes for its states, it should... (Except you have to measure, not count, those "N" states. It's the same sort of problem as saying a line containing "N points", which is sorta misleading but I don't know a better wording.)

> [Wallace] has to assume branching indifference, and that, it turns out, is where the Born rule is hiding.

No, as I explained previously branching indifference is trivial.

> The only reason there was a problem in the first place is because a photon was taken (on good evidence) to be an indivisible unit that could not "split up". So which is it? Can a photon split up or not?

Single-universe photons do not split. Multiversal photons do not split but are already pre-"split" in a way very similar to how a 2-inch line segment is already pre-"split" into points instead of being an indivisible whole.

> No, I mean that "the agent" can refer to either the pre-split agent, or any one of the N post-split agents. If you're going to talk about *the* agent (singular) then you have to specify which of the N+1 agents you mean.

The agent at the start of the scenario is a collection of N fungible (identical) agents.

There is no splitting, ever New agents are never created. There is only differentiating: agents that already existed, and were formly identical, become different.

> Dealing with "minor fluctuations" often requires a radical departure from the conventional wisdom.

A tiny fraction of minor fluctuations turn out not to be minor. Most really are minor, e.g. if you measure a length, weight or temperature of a glass of water there will be tiny fluctuations (of the glass, the water, and the measuring instrument) that affect your measurement but they are usually too minor for you to even notice.

> If a multiversal agent in N states tries to maximize the outcomes for its states, it should... (Except you have to measure, not count, those "N" states. It's the same sort of problem as saying a line containing "N points", which is sorta misleading but I don't know a better wording.)

What about from the perspective of an individual agent instance? That's a bit of an approximation because what really exists are multiversal objects, not single universe objects. But it works pretty OK. From that perspective, basically what happens is probability: if 2/3 of the instances in the multiversal object get outcome X then from the perspective of a single instance it has a 2/3 chance to get outcome X.

> You don't actually count universes, you measure them, just like you don't count points, you measure them.

Ah. How is that done exactly? If I wanted to "measure" the universe I inhabit, how would I do it? If I measured my universe before and after a quantum event, would I get a smaller result the second time?

> there is nothing about MWI which contradicts any of your subjective experiences

Of course there is. MWI says that there are many of me (that's what the M in MWI stands for!) But my subjective experience is that there is only one of me.

> you have to measure, not count, those "N" states

And what exactly compels me to measure and not count? Why am I compelled to use the measure of a universe to make decisions rather than coalescing all of the fungible versions of me into one equivalence class and treating that as a single entity?

> branching indifference is trivial

Being trivial and being the root of the begging of the question are not mutually exclusive.

> Single-universe photons do not split.

Wait, what? What is a "single-universe photon"? Just now you were telling me that "you don't actually count universes, you measure them" and that this is the reason that universes can split/differentiate/whatever-you-want-to-call-it forever. So what on earth can a "single-universe photon" possibly mean?

Are you familiar with single-photon sources?

https://en.wikipedia.org/wiki/Single-photon_source

Do those emit "single-universe photons" that "do not split"? If so, what exactly happens if I run one of those through a beam splitter?

> The agent at the start of the scenario is a collection of N fungible (identical) agents.

Is N an integer?

> > Dealing with "minor fluctuations" often requires a radical departure from the conventional wisdom.

> A tiny fraction of minor fluctuations turn out not to be minor. Most really are minor, e.g. if you measure a length, weight or temperature of a glass of water there will be tiny fluctuations (of the glass, the water, and the measuring instrument) that affect your measurement but they are usually too minor for you to even notice.

For day-to-day life that is certainly true. But we're talking about quantum physics here. In that realm, tiny fluctuations matter *a lot*.

Peter asked you a question about the mathematics of MWI and you answered by referring him to a text that contains no math. I can think of only two possible explanations:

1. You thought Peter was a layman and referred him to a popular text because you thought that would be appropriate.

2. You knew Peter was not a layman, but referred him to a popular text nonetheless for reasons that I cannot even begin to guess. (Surely it wasn't because you were unaware of the Deutsch paper. You know the MWI literature better than I do.)

So I guessed #1 to be the case and decided to give you the heads-up. That's all.

> Being trivial and being the root of the begging of the question are not mutually exclusive.

If you have some argument about why branching indifference is false, or has some other problem in this context, please explain it. As far as I could tell, your objections to it came from not understanding it (or, in the alternative, understanding it differently than I do). So I explained what I thought it meant. IIRC you seemed to agree with me. But now you bring it up again as problematic.

> Ah. How is that done exactly? If I wanted to "measure" the universe I inhabit, how would I do it? If I measured my universe before and after a quantum event, would I get a smaller result the second time?

Measures can be abstract or hypothetical. They aren't only things you can actually measure. The concept of a measure does not require a physical process does performs that measure. Here is some explanation of measures, and relates issues, based on conversations with Deutsch (he taught it to me):

That is not a full explanation of everything. It's just a starting point. If you agree with what I've said so far, in my comment and a the link, I'll continue from there. If there's a disagreement with this part, then we can talk about that first.

> > there is nothing about MWI which contradicts any of your subjective experiences

> Of course there is. MWI says that there are many of me (that's what the M in MWI stands for!) But my subjective experience is that there is only one of me.

You have never in your life observed the absence of other instances of you in the multiverse. You have looked only at a limited portion of the multiverse and didn't see more of you. That is fully compatible with MWI.

Your experience, taken more literally, is e.g. that you looked down and saw your foot. Observations like that are valid data but do not contradict MWI. You seem to be mixing up your personal (= made by you instead of someone else) observations with vague feelings which are not really obsrvation data but are intellectual intuitions.

> And what exactly compels me to measure and not count? Why am I compelled to use the measure of a universe to make decisions rather than coalescing all of the fungible versions of me into one equivalence class and treating that as a single entity?

It's like the difference between 1 inch and 2 inches. If you just say "a linear set of points" and treat all such sets as equivalent, you will deal with distance poorly.

I have read too many pop science books by physicists that misrepresent the actual physics, so I don't trust pop science books to properly present the physics. Since I'm perfectly capable of reading the actual papers and understanding the actual math, that's what I do.

(It's possible that the pop science book you refer to is an exception and actually does properly present the physics. But the only way I can know that is to look at the actual physics, i.e., the papers and the math. So it's still a waste of time for me to read the pop science book.)

I'll do some looking at the literature and come back if I have further questions.

> I would refer anyone to BoI. I think it's the best thing to read first to understand this

BoI is a pretty good intro for a layman (though I think TFoR was better). But it's inappropriate for anyone seeking technical detail. It doesn't connect the intuitions to the math at all, and it puts way too much emphasis on the transporter metaphor IMHO.

> why branching indifference is false

Branching indifference is not false. What is false is the proposition that an agent must accept branching indifference in order to be considered rational. That turns out to be the Born rule in disguise.

To illustrate how a plausibly rational agent could reject branching indifference (this example is due to Tim Maudlin, though I've reframed it here): suppose I am offered a choice between chocolate or vanilla ice cream. I like them both, and would prefer to have both, but I can't. I have to choose (maybe I only have enough money for one scoop). Instead of choosing according to my free will, I instead perform a quantum experiment in order to split myself into two. One of me chooses chocolate, the other chooses vanilla. A rational agent could plausibly prefer this state of affairs (assuming they care about the multiverse *at all*) to one where only one flavor is actually chosen.

A more fundamental problem is that you cannot a priori rule out branch counting as a plausibly rational strategy. Wallace argues against it by retreating from the proposition that "branch" is a meaningful concept (thereby undermining all of the popular rhetoric of the MWI, but let's leave that aside) but this argument fails because I can just *define* some way of coalescing slices of the universe into equivalence classes which I call "branches", and then I can count them. Wallace concedes this: "...within the stylised context of my decision theory, the branch count is defined." And then, remarkably, he actually goes on to (tacitly) concede that he has begged the question! "... so of course (given the representation theorem) the branch counting rule must violate some of my axioms. In fact, it violates ... branching indifference ..."

> Measures can be abstract or hypothetical. They aren't only things you can actually measure.

Yes, I understand that. (You are either severely underestimating my grasp of the basics of the MWI, or I am overestimating yours.) My question is: is the "measure" of the universe I inhabit something I can actually measure? (Answer: no, it isn't. You knew that right?)

> You seem to be mixing up your personal (= made by you instead of someone else) observations with vague feelings which are not really observation data but are intellectual intuitions.

No. I accept, for example, that black holes exist despite the fact that I have never personally experienced one. The difference is that *in principle* (and nowadays in actual practice) there are experiments one can do to demonstrate the existence of black holes. But the MWI says that the existence of other versions of me is not experimentally demonstrable (to me) *even in principle*, not even with arbitrarily advanced technology. That's what makes the MWI untenable.

> The concept of a measure does not require a physical process does performs that measure.

It does if you want me to take it seriously as something that makes contact with reality. If I can't measure it, not even in principle (and I can't) then It's an IPU -- an Invisible Pink Unicorn -- no different than particle positions in the Bohm interpretation. (In fact, if you work out the math, branch weights play *exactly* the same role in the MWI as particle positions play in Bohmian mechanics: they are the IPU, the unmeasurable-even-in-principle quantity, where the Bornian randomness is carefully hidden.)

> It's like the difference between 1 inch and 2 inches.

The difference being: I can demonstrate a physical process that allows me to measure and compare distances in a meaningful and logically coherent way. (This is quite a remarkable fact about the universe, BTW.) You cannot do the same for the measure of a universe, not even in principle.

> Branching indifference is not false. What is false is the proposition that an agent must accept branching indifference in order to be considered rational. That turns out to be the Born rule in disguise.>> To illustrate how a plausibly rational agent could reject branching indifference (this example is due to Tim Maudlin, though I've reframed it here): suppose I am offered a choice between chocolate or vanilla ice cream. I like them both, and would prefer to have both, but I can't. I have to choose (maybe I only have enough money for one scoop). Instead of choosing according to my free will, I instead perform a quantum experiment in order to split myself into two. One of me chooses chocolate, the other chooses vanilla. A rational agent could plausibly prefer this state of affairs (assuming they care about the multiverse *at all*) to one where only one flavor is actually chosen.

What is the sense in which this is supposed to be rational? How would one enact this rule in a way that's consistent with quantum mechanics, having consistent preferences over time and so on?

Wallace and Deutsch have given answers to those questions. You haven't AFAIK.

>> It's like the difference between 1 inch and 2 inches.>> The difference being: I can demonstrate a physical process that allows me to measure and compare distances in a meaningful and logically coherent way. (This is quite a remarkable fact about the universe, BTW.) You cannot do the same for the measure of a universe, not even in principle.

You can measure the measure of a universe:

https://arxiv.org/abs/1108.5329

More generally, any experiment you conduct to test quantum mechanics has to take into account the amplitude of different possible outcomes. This includes experiments in which macroscopic objects exist in multiple versions. You have offered no alternative explanation.

> > Branching indifference is not false. What is false is the proposition that an agent must accept branching indifference in order to be considered rational. That turns out to be the Born rule in disguise.> >> > To illustrate how a plausibly rational agent could reject branching indifference (this example is due to Tim Maudlin, though I've reframed it here): suppose I am offered a choice between chocolate or vanilla ice cream. I like them both, and would prefer to have both, but I can't. I have to choose (maybe I only have enough money for one scoop). Instead of choosing according to my free will, I instead perform a quantum experiment in order to split myself into two. One of me chooses chocolate, the other chooses vanilla. A rational agent could plausibly prefer this state of affairs (assuming they care about the multiverse *at all*) to one where only one flavor is actually chosen.> > What is the sense in which this is supposed to be rational? How would one enact this rule in a way that's consistent with quantum mechanics, having consistent preferences over time and so on?

Ron, branching indifference is not directly related to the Born rule, let alone equivalent. I think you must be combining it with some other ideas to reach something like a Born rule equivalent. You think that, along with some other premises, and via some reasoning, it *implies* the born rule. Right? If you disagree, please define branching indifference, define the born rule, and then point out the equivalence. If I'm right, please actually provide the reasoning involved.

Regarding the ice cream, I have a guess at what Ron might have in mind. Suppose I flip a coin so that I get each flavor in ~half of the universes. Then I can eat chocolate in a particular universe and think to myself "I would have liked to have both flavors; but I don't have them; but I know that, right now, versions of me are eating vanilla and that thought brings me a portion fo the satisfaction i would have gotten from personally having a half-portion of vanilla."

I think what's going on is the agent prefers to eat chocolate *and to have this thought* than to eat chocolate alone. In other words, the agent gets a higher reward, and more satisfaction from the ice cream, due to certain thoughts and the coin flipping action (his satisfaction from these thoughts depends on his belief this is really happening, and the coin flip enables that so that he isn't worried that he chose chocolate in ~all universes).

In this scenario, we're comparing alternatives with unequal rewards. So that is why one is preferred. This does not contradict branch indifference which is just saying that agents don't care about branching *indpendent of* any change in rewards.

If you drop the thought process about the multiverse from the scenario, then what is happening in those other universes cannot be relevant to the satisfaction/reward for an agent in a particular universe. That is, consider 2 hypothetical agents in separate multiverses. These are different scenarios. One eats chocolate and so do all his clones. The other eats chocolate but half his clones eat vanilla. Neither agent has any thoughts about what his multiversal clones are doing. Everything else being equal, the rewards are equal – rewards cannot depend on stuff you don't know about which never affects you in the future.

In the sense that Deutsch defines in his original paper: "‘Rationality’ ... means conformity to a set of constraints on a decision maker’s preferences."

> How would one enact this rule in a way that's consistent with quantum mechanics, having consistent preferences over time and so on?

Quantum mechanics does not require a rational agent to have consistent preferences over time, what Wallace calls diachronic consistency. That is another assumption that Wallace and Deutch have to make in order to achieve their result. I could just as well have challenged that as branching indifference, but decided not to for the sake of brevity. Nonetheless, diachronic consistency is clearly not necessary to be rational. Rational agents can change their preferences, e.g. on the basis of new information. If diachronic consistency were required for rationality, no human being could ever be rational.

But I'd really prefer not to get into those weeds.

> Wallace and Deutsch have given answers to those questions. You haven't AFAIK.

I just did.

> You can measure the measure of a universe:

That paper does not support your claim. The word "universe" doesn't even appear in it, nor does it appear on the Wikipedia page for "quantum tomography."

You're going to have a very hard time explaining to me how you can measure the measure of a universe in light of the fact that "universe" is not actually a well-defined term under the MWI. But that's kind of beside the point because the question I actually asked was not "can I measure the measure of *a* universe" but rather "Can *I* measure the measure of *the* universe that *I* inhabit?" And the answer to that question is: no (because that would allow me to violate the no-cloning theorem).

> You think that, along with some other premises, and via some reasoning, it *implies* the born rule. Right?

Well, yeah, of course. You have just described David Wallace's paper.

I *accept* the argument that Wallace presents, i.e. I accept that the conclusion follows from the premises. But I reject (some of) the premises. Specifically, I reject branching indifference as a precondition on rationality. (I also reject diachronic consistency as a precondition on rationality, but let's focus on one thing at a time.)

Branching indifference is equivalent to the Born rule in the same sense that Euclid's parallel postulate is equivalent to (for example) the proposition that the sum of the internal angles of a triangle equals the sum of two right angles.

> Specifically, I reject branching indifference as a precondition on rationality.

Oh, you *reject branching indifference as a precondition on rationality*. Either you didn't say it quite like that before (with the "as" part) or I missed it. That makes more sense to me.

So, do you accept that that branching indifference fits with a standard game theory view of a "rational agent"? Or do you deny that too? I think that's the relevant issue because the goal is not talking about rationality in general, just to see what conclusions one can reach based on non-probabilistic parts of QM and some game theory premises.

And a good next step would be to give a counter example (a rational agent violating branching indifference), although if you have a different way of arguing your point that'd be OK too. I think the ice cream scenario was intended to be a counter example, but I think I've answered that one.

> Oh, you *reject branching indifference as a precondition on rationality*. Either you didn't say it quite like that before (with the "as" part) or I missed it. That makes more sense to me.

I didn't say it because I thought it was obvious. The only reason we're talking about "branching indifference" *at all* is that David Wallace introduces it as an assumption in the paper I cited. In that paper, "branching indifference" is an assumption introduced in a section entitled "The dictates of rationality" and specifically introduced by a paragraph beginning with the phrase, "The remaining rationality axioms..." I assumed you knew all that.

> So, do you accept that that branching indifference fits with a standard game theory view of a "rational agent"?

That depends on what you mean by "fits". A rational agent certainly *can* adopt branching indifference. I dispute that a rational agent should be *required* to adopt it in order to be considered rational. (That's actually a pretty easy case to make: I believe that an agent could plausibly be rational in a purely classical world, i.e. without accepting quantum mechanics *at all*. So requiring the acceptance of branching indifference aaPoR is a lot to ask.)

> I think the ice cream scenario was intended to be a counter example, but I think I've answered that one.

Yes, and your restatement of my position was exactly right (except for the minor quibble that a regular coin flip does not suffice. You have to do a *quantum* coin flip.)

The problem with your counter-argument is this:

> If you drop the thought process about the multiverse from the scenario

That is a very peculiar counterfactual for a proponent of MWI to raise. If you "drop the thought process about the multiverse from the scenario", then the entire argument completely falls apart. You can't have it both ways: either the evidence logically compels the belief that MWI is true, or it does not. If the evidence compels the belief that MWI is true, then a rational agent who is aware of the evidence cannot simply decide to ignore this and act as if it weren't true. *That* really is a compulsory part of what it means to be rational.

> (That's actually a pretty easy case to make: I believe that an agent could plausibly be rational in a purely classical world, i.e. without accepting quantum mechanics *at all*. So requiring the acceptance of branching indifference aaPoR is a lot to ask.)

I don't think it's a branching indifference violation to live in a classical world. It makes sense to be indifferent to impossibilities rather than to have preferences about them.

> That is a very peculiar counterfactual for a proponent of MWI to raise. If you "drop the thought process about the multiverse from the scenario", then the entire argument completely falls apart. You can't have it both ways: either the evidence logically compels the belief that MWI is true, or it does not. If the evidence compels the belief that MWI is true, then a rational agent who is aware of the evidence cannot simply decide to ignore this and act as if it weren't true. *That* really is a compulsory part of what it means to be rational.

If an agent is unaware of the multiverse, or not currently thinking about it, then *for many isolated scenarios*, a classical analysis is OK.

If the agent is thinking about the multiverse, then the state of the multiverse (as known to that agent's thoughts) is relevant to the agent's preferences and rewards, so there is no conflict with branching indifference. Of course, the agent may still be indifferent to the particular branching in question even if he considers it. And, of course, the state of the multiverse will also be relevant, whether the agent thinks about it or not, if it actually interacts with the agent ever again.

> I don't think it's a branching indifference violation to live in a classical world.

> If an agent is unaware of the multiverse

You appear to have lost the plot here.

This entire discussion is about the claim that the Born rule can be derived from the unitary dynamics of QM plus decision theory. Specifically, the claim is (quoting Wallace):

"... the only rational strategy for an agent in an Everettian universe is to follow the Born rule."

This is true (or at least it has been proven) only if it is necessary for an agent who knows they are living in a multiverse to subscribe to branching indifference (and diachronic consistency) in order to be considered rational. It it were possible to prove this to be the case, Wallace would have proved it instead of adopting this as axioms. (I suppose it's possible that these can be proved but neither Wallace nor Deutsch are clever enough to figure out how to do it. I hope we can agree to discount this possibility.)

So they are axioms, and from these axioms the conclusions do indeed follow. The problem is that neither of these axioms is self-evidently true, at least not to me. In fact, the necessity of branching indifference for rationality for an agent that knows it is living in an Everettian universe seems self-evidently false to me, as illustrated by the ice-cream example. Unless you can explain to me why my preference for ice cream flavor variety in the multiverse is necessarily irrational, the argument fails.

Here's another way to look at it. Recall this from the explanation of why branch counting is irrational:

"...within the stylised context of my decision theory, the branch count is defined, so of course (given the representation theorem) the branch count- ing rule must violate some of my axioms. In fact, it violates the combination of branching indifference and diachronic consistency."

But I do not accept the necessity of branching indifference and diachronic consistency for rationality, so I do not accept the irrationality of branch counting. Branch counting, within the "stylised context of [Wallace's] decision theory" seems eminently rational to me. Why is it not rational to desire to maximize the reward for the greatest number of distinct agents? That's a perfectly admissibly objective function for a rational agent in a single universe, so why not for the multiverse? If I, say, win the lottery, why should I not rationally prefer to split myself as much as possible in order to create as many copies/versions of myself as possible, all of which can enjoy copies (and in this case they really are copies, not versions) of my lottery winnings?

And remember that you have to answer this question without reference to weights because that too is sneaking the Born rule in through the back door.

Then when I restated part of my own restatement, talking about the same thing again, instead of it being "exactly right" you decided I'd lost the plot. I don't think you understood either of my messages.

Now, after claiming my refutation of your ice cream example was "exactly right", you claim you've already won the argument via the ice cream example. You are lost.

> And remember that you have to answer this question without reference to weights because that too is sneaking the Born rule in through the back door.

"Weights" are a fundamental, non-probabilistic part of QM which can be legitimately referenced. I think the biggest is that you are not on board with DD's premises about QM. So let's try this. Tell me which of the following quotes you think are false:

https://arxiv.org/pdf/quant-ph/0104033.pdf

> In other words, when such sub-networks are in identical states, they are *fungible*. The term is borrowed from law, where it refers to objects, such as banknotes, that are deemed identical for the purpose of meeting legal obligations. In physics we may define entities as fungible if they are not merely deemed identical but *are* identical, in the sense that although they can be present in a physical system in varying numbers or amounts, permuting them does not change the physical state of that system. Fungibility is not new to physics. Many physical entities, such as amounts of energy, are fungible even in classical physics: one can add a Joule of energy to a physical system, but one cannot later extract the same Joule.

> A multiset is like a set except that some of its elements are fungible. Each element is associated with an integer, its *multiplicity*, which specifies how many instances of it appear in the multiset.

> An ensemble is a limiting case of a multiset where the total number of elements M goes to infinity but all the proportions [...] tend to definite limits.

> I shall refer to a non-empty sub-ensemble in which all the computers are in the same state as a branch of the ensemble

> These properties give each branch a well-defined identity over time, even though the values of its bits change.

> There are such things as fungible processes as well as fungible objects.

> The effect of an *n*-qubit quantum gate during one computational step is to transform the *3n* matrices representing the *n* participating qubits into functions of each other in such a way that the relations (14) are preserved.

> in order to model information flow we are using local interactions (gates) of the network to model local interactions in general quantum systems

> Given the universality of the Toffoli gate, all these properties must hold whenever a quantum network, or any part of it, performs a classical computation. In other words, whenever any quantum network (including a sub-network of another network) is performing a classical computation f, the matrices [...] for that network evolve independently of all its other descriptors.

> Thus in any sub-network *R* of a quantum computational network where a reversible classical computation is under way, half the parameters describing *R* are precisely the descriptors of an ensemble of classical networks. It is half the parameters because, from (14), any two of the three components [...] determine the third. This does not imply that such a subsystem constitutes half the region of the multiverse in which *R* exists. Proportions in the latter sense – which formally play the role of probabilities under some circumstances, as shown in Deutsch (1999) – are determined by the Heisenberg state as well as the observables, and do not concern us here because the present discussion is not quantitative. >> The other half of the parameters, [...] contain information that is physically present in *R *(it can affect subsequent measurements performed on *R *alone) but cannot reach the ensemble (the descriptors of the ensemble being independent of that information). But the reverse is not true: as (19) shows, information can reach the quantum degrees of freedom from the ensemble.

> The proposition that parts of the multiverse have the same description as an ensemble with given properties is not quite the same as the proposition that such an ensemble is actually present in those parts of the multiverse, for the description might refer to entities that are not present in addition to those that are. In particular, an ensemble has an alternative interpretation as a *notional *collection, only one member of which is physically real, with the multiplicity of a given branch representing the probability that the properties of that branch were the ones prepared in the real system at the outset, by some stochastic process. However, no such interpretation is possible if the branches affect each other, as they do in general quantum phenomena, and in quantum computations in particular (see Benjamin 2001).

> When a quantum computational network is performing a general computation, it need not be the case that the descriptors of any part of the network over two or more computational steps constitute a representation of an evolving e-algebra. [...] so the conditions discussed in Section 3 for branches to have an identity over time need not hold.

> In a typical quantum algorithm, [...] the qubits first undergo a non-classical unitary transformation [...], then a reversible classical computation, and finally another unitary transformation which is often the inverse [...] of the first one. Despite the fact that the branches lose their separate identities during the periods of the quantum transformations [...] we can still track the flow of information reasonably well in terms of ensembles:

> Therefore, if some sub-network of a quantum network performs a classical computation for a period if the network is isolated, and then it is run with some or all of the observables [...] being repeatedly measured between computational steps, it will still perform the same classical computation and will contain an ensemble identical to that which it would contain if it were isolated (though its other descriptors will be very different).

> Since a generic quantum computational network does not perform anything like a classical computation on a substantial proportion of its qubits for many computational steps, it may seem that when we extend the above conclusions to the multiverse at large, we should expect parallelism (ensemble-like systems) to be confined to spatially and temporally small, scattered pockets. The reason why these systems in fact extend over the whole of spacetime with the *exception *of some small regions (such as the interiors of atoms and quantum computers), and why they approximately obey classical laws of physics, is studied in the theory of decoherence (see Zurek 1981, Hartle 1991). For present purposes, note only that although most of the descriptors of physical systems throughout spacetime do not obey anything like classical physics, the ones that do, form a system that, to a good approximation, is not only causally autonomous but can store information for extended periods and carry it over great distances. It is therefore that system which is most easily accessible to our senses – indeed, it includes all the information processing performed by our sense organs and brains. It has the approximate structure of a classical ensemble comprising ‘the universe’ that we subjectively perceive and participate in, and other ‘parallel’ universes.

>> What is the sense in which this is supposed to be rational?>> In the sense that Deutsch defines in his original paper: "‘Rationality’ ... means conformity to a set of constraints on a decision maker’s preferences."

DD didn't pick constraints arbitrarily.

>> How would one enact this rule in a way that's consistent with quantum mechanics, having consistent preferences over time and so on?>> Quantum mechanics does not require a rational agent to have consistent preferences over time, what Wallace calls diachronic consistency. That is another assumption that Wallace and Deutch have to make in order to achieve their result. I could just as well have challenged that as branching indifference, but decided not to for the sake of brevity. Nonetheless, diachronic consistency is clearly not necessary to be rational. Rational agents can change their preferences, e.g. on the basis of new information. If diachronic consistency were required for rationality, no human being could ever be rational.

If you change your priorities then the way you value outcomes will change. But without diachronic consistency it's impossible for you to enact preferences consistently regardless of whether you change your mind.

> But I'd really prefer not to get into those weeds.

I'm puzzled. Do you want to discuss substantive issues or not? Are you just wasting your time and mine?

>> Wallace and Deutsch have given answers to those questions. You haven't AFAIK.>> I just did.

Above you denied that consistent decisions are possible because you might change your mind about something. Now you're saying you provided a consistent rule for rational decision making. Also, you haven't provided a detailed discussion of how one would go about enacting your rule, so you haven't provided answers.

>> You can measure the measure of a universe:>> That paper does not support your claim. The word "universe" doesn't even appear in it, nor does it appear on the Wikipedia page for "quantum tomography.">> You're going to have a very hard time explaining to me how you can measure the measure of a universe in light of the fact that "universe" is not actually a well-defined term under the MWI.

Universes are an approximation. If I pick some particular approximation and stick to it then there's no reason I can't discuss the measure of a universe.

> But that's kind of beside the point because the question I actually asked was not "can I measure the measure of *a* universe" but rather "Can *I* measure the measure of *the* universe that *I* inhabit?" And the answer to that question is: no (because that would allow me to violate the no-cloning theorem).

You can do experiments to test the evolution of the amplitudes of different possible outcomes a system over time. This doesn't require measuring the measure of the system you're currently in over the entire multiverse.

No, that's not what I said. I said that your re-statement of my example was (almost) exactly right, i.e. this part:

> Suppose I flip a coin so that I get each flavor in ~half of the universes. Then I can eat chocolate in a particular universe and think to myself "I would have liked to have both flavors; but I don't have them; but I know that, right now, versions of me are eating vanilla and that thought brings me a portion fo the satisfaction i would have gotten from personally having a half-portion of vanilla."

I don't accept your subsequent refutation because...

> Then when I restated part of my own restatement, talking about the same thing again, instead of it being "exactly right" you decided I'd lost the plot.

Yes, that's right, because you started talking about agents who do not accept MWI. It's not that what you said about these agents was *wrong*, it's that talking about agents who do not accept MWI is *irrelevant* here (except as it informs the choice of criteria for what should be considered rational in general) for the reasons that I explained in my previous comment.

> "Weights" are a fundamental, non-probabilistic part of QM which can be legitimately referenced.

Yes, of course. But you have to remember that the only way we can ever actually know the weight of a quantum system is if it has been *prepared*. We cannot know the weight of an unprepared system because of the no-cloning theorem. So yes, weights can be legitimately referenced. It does not follow that "the weight of a universe" can be legitimately referenced because universes cannot be prepared.

> Tell me which of the following quotes you think are false:

None of them are false. But I don't see why you think any of them are relevant to a discussion of whether or not branching indifference should be assumed to be a precondition for rationality.

Of course he didn't. He chose constraints that lead to the result he wanted to demonstrate. (BTW, Deutsch never mentions branching indifference at all. In his paper, it's a tacit assumption, and one that is by no means easy to discern. Only in the Wallace paper is it explicitly discussed.)

> > But I'd really prefer not to get into those weeds.

> I'm puzzled. Do you want to discuss substantive issues or not? Are you just wasting your time and mine?

I would prefer to deal with one issue at a time. When we have resolved the question of whether branching indifference should be assumed to be a precondition for rationality then we can go on, if you wish, to examine the question of whether diachronic consistency should likewise be assumed.

> Universes are an approximation. If I pick some particular approximation and stick to it then there's no reason I can't discuss the measure of a universe.

Of course you can *discuss* it. The question is whether "the measure of a universe" corresponds to something physically real, or whether it's an artificial mathematical construct, like Bohmian positions, inside of which Bornian probabilities have been cleverly hidden. I claim it's the latter. And my support for this is that the measure of a universe, like Bohmian positions, cannot ever be known to the inhabitants of that universe even in principle.

> You can do experiments to test the evolution of the amplitudes of different possible outcomes a system over time.

No, you can't. Once you do an experiment on a quantum system you *change* it in such a way that you cannot go back and do another experiment on the same system. Specifically, to perform an experiment on a system you have to entangle that system with a measurement apparatus in a thermodynamically irreversible way.

The best you can do is perform the same experiment on a *plurality* of systems that have been prepared in (presumably) identical states and then go back and retroactively try to tell a story to explain the behavior that you observe. It turns out that when you do this, two interesting things happen:

1. The outcomes of your experiments will not always yield the same results despite your best efforts to prepare the systems in identical states.

2. The only way to describe the prepared state of a quantum system in general that is consistent with the evidence is with a function whose domain is phase space and whose range is complex numbers.

That's really it. All the rest, including "particles" and "universes", is dramatic narrative.

"... the wave function is not an objectively real object. Instead, it is a collection of numbers describing the observer's knowledge about the world."

He doesn't get around to this until the very last paragraph.

Well, Lubos, if this "collection of numbers" describes someone's "knowledge about the world", what exactly is the nature of this thing of which the possessor of this collection of numbers has knowledge? Particularly in light of the fact that these are *complex* numbers we're talking about here. (This is usually the point at which Copenhagenists say, "Shut up and calculate.")

(Just for the record, Lubos's statement is actually wrong, or, at best, an oversimplification. The wave function is not a "collection of numbers", it's a *function*, and in particular, it's a function on *phase space*, not physical space. If this function represents "knowledge", what exactly is the nature of the thing that it is knowledge *of*? Because it ain't classical reality.)

One of the frustrating things about this whole QM interpretation debacle is that *everyone* is wrong about *something* (except me, of course. I understand it all perfectly, but I seem to be the only one ;-)

He doesn't get around to this until the very last paragraph.Not quite. In the middle of his post he says:

"The observer-dependence may be seen e.g. in the Wigner's friend setup. Wigner observes a friend who observes an elementary particle. They will unavoidably have a different description because the friend collapses the wave function after his measurement, while Wigner hasn't made any measurement by that time, and describes the whole lab as an entangled state of the friend and the particle. So Wigner and his friend clearly use a different wave function."

Is the "wave function" objectively real, or not? Lubos says it isn't, MWI says that it is.

what exactly is the nature of this thing of which the possessor of this collection of numbers has knowledgeWe don't know. All we know is how it behaves, not what it really is. At least, according to Feynman. Do you know what it is?

*everyone* is wrong about *something* Other than Lubos' oversimplification, what is he wrong about in his post?

> Other than Lubos' oversimplification, what is he wrong about in his post?

The fundamental problem that *everyone* makes is insisting that existence is a binary quality. A thing either exists, or it does not. This seems like a plausible assumption, but it is in fact false, and so if you base your reasoning on it you will go seriously astray. And it's not just physicists. Theologians get this wrong too. (This may be one reason why arguing with hard-core many-worlders like Elliot and Alan often feels very similar to me to arguing with YECs.)

It is simply not the case that the wave function either exists or does not exist. The wave function is simply in an ontological category of its own, a category that is distinct from the ontological category of rocks (classical objects), which is in turn distinct from the ontological category of Darth Vader (fictional characters), which are all distinct from the ontological category of Invisible Pink Unicorns (unfalsifiable things) and four-sided triangles (logically impossible things). And even that is nowhere close to an exhaustive list of all the ontological categories that exist (!).

@Ron:The wave function is not a "collection of numbers", it's a *function*, and in particular, it's a function on *phase space*, not physical space.

I don't think the "function on phase space" part is correct. The wave function is a function on the configuration space of the system, but that's not the same as phase space.

For example, consider two spinless particles each moving along one dimension, which is the simplest case for which the wave function cannot be viewed as a simple function on physical space. The wave function is a function of either two positions (in the position representation) or two momenta (in the momentum representation). Those are functions on (representations of) configuration space, which is a two-dimensional space because there are two particles each with one degree of freedom (adding spin or more spatial dimensions would add more degrees of freedom per particle). But the *phase* space of the system is a *four* dimensional space, since it has a dimension for the position and momentum of each particle.

It may come as a surprise to you, but corresponding on this blog is not the only thing I do day to day. I have a life. And when someone asks me a question that includes a word with a well-established technical meaning, I can sometimes forget that that word has been locally redefined earlier in the conversation, especially after some time has passed.

So now that I know what you're talking about, sure, the multiverse is a Deutschian-ensemble (let's abbreviate that D-ensemble to distinguish it from the more usual meaning of statistical ensemble). So? I *still* don't see why you think this is relevant to a discussion of whether or not branching indifference should be assumed to be a precondition for rationality.

It is your responsibility to take into account the context of a discussion such as my immediately prior message and, more broadly, the literature under discussion which you claim to have deep dived on and to have understood in detail. You failed at your responsibility of taking into account discussion context and are now belligerent about your error. That is unreasonable and is representative of the general carelessness with which you approach intellectual matters.

You have still neglected to actually answer my question. The multiverse is an ensemble *of what* with *what multiplicities*?

The reason I'm talking about the multiverse in general is because we disagree about it (or understand it differently, or something) and that disagreement is a major underlying factor in the probability discussion. It comes up in e.g. your claims about branch counting.

Funny, I don't remember entering into any kind of contractual relationship with you that obligates either of us in any way.

> You failed...

Yes, I did. I'm a fallible human being. If you want to interact with someone who never fails you will need to go elsewhere.

> and are now belligerent

Yes, I'm a crotchety old man, and I sometimes get belligerent when I'm dealing with self-righteous twits. It's another failing of mine.

> The multiverse is an ensemble *of what* with *what multiplicities*?

I have no idea, and I don't see how this can possibly shed any light on the matter at hand. If you want to explain why you think that branching indifference should be assumed as a pre-requisite for rationality, or otherwise defend your position, by all means go for it. But I'm not playing your stupid Socratic game.

To summarize, there are three varieties of out-of-body experience in the limited medical literature on the subject, in one of which a person not only experiences themselves as here, but also as across the room. Drives people crazy.

Point being that the experience of self, or what we are all familiar with as the experience of self, depends on the coordination of the vestibular, proprioceptive, graviceptive, and ocular senses. If any of these senses are dysfunctional, out-of-body experience can result.

Thanks for your description of the physics. It's mostly Greek to me, but fascinating.

Ron said...>Well, Lubos, if this "collection of numbers" describes someone's "knowledge about the world", what exactly is the nature of this thing of which the possessor of this collection of numbers has knowledge? Particularly in light of the fact that these are *complex* numbers we're talking about here.

>(Just for the record, Lubos's statement is actually wrong, or, at best, an oversimplification. The wave function is not a "collection of numbers", it's a *function*, and in particular, it's a function on *phase space*, not physical space. If this function represents "knowledge", what exactly is the nature of the thing that it is knowledge *of*? Because it ain't classical reality.)

Obviously I'm not Lubos, but he is correct. OK, so it's the *State Vector* that's just a (complex) number, and the wave *function* is a function that predicts the evolution of the State Vector with time.

What is the nature of the underlying reality? Sorry, that's a (possibly meaningless) philosophy question.

What kind of "knowledge" does the wave function represent? Just possibilities. The only difference from classical probability is that in QM these possibilities can interfere with each other. Thus the possibility that a particle could have gone thru one slit interferes with the the possibility it went thru the other. That's why it's a function in configuration space. Here just read 'possibility' as 'amplitude of the wave function'.

That's just superposition, which can be explained by local action.

But Psi is not the wave function of a particle. Psi is defined in Quantum Mechanics as the wave function of the system. And that means Psi cannot be entirely local. It is this feature of QM that gives rise to Entagelment.

A *collection* of complex numbers. (Precision matters if you're going to talk about who is right and who is wrong.) But even that is misleading because that collection is not an unstructured set. The state vector is a function whose range is complex numbers and whose domain is the configuration space of a system. The wave function simply adds time as another dimension of the domain, so these two terms are often safely interchangeable in many contexts.

> What is the nature of the underlying reality? Sorry, that's a (possibly meaningless) philosophy question.

Well, of course it's a philosophy question! But saying that it's meaningless is just throwing in the towel. If *you* want to throw in the towel, that's perfectly fine, but unless you can *prove* that it's a meaningless question it is not fair to cast aspersions on those who are unwilling to follow your lead and just shut up and calculate.

> What kind of "knowledge" does the wave function represent? Just possibilities.

The "just" part of this claim requires further justification. Yes, obviously the wave function represents possibilities. But *just* possibilities? How do you know? How can you be certain that it isn't a reflection of some kind of physical *mechanism* which can be described in terms that are more enlightening than the bare math in the same way that "curved space-time" is more enlightening (to most people) than the Einstein field equations?

> The only difference from classical probability is that in QM these possibilities can interfere with each other.

@RonYes, I could have taken precision to the point of making my comment unreadable. I chose not to, as it wouldn't mater. We both know what a vector is.

"shut up and calculate": No, I am not in that crowd. Amazing you could jump to that conclusion based on a single sentence. Perhaps you could hold an open mind just a little longer? Maybe long enough to see the word "possibly" in that sentence?

I simply see no reason the equations of Quantum Mechanics should answer such philosophical questions. Why not look to General Relativity for such answers instead? Or any other physics theory? (That's not a rhetorical question. I'm seriously asking why should you (if you do) think that QM of all physics has answers to questions in philosophy?)

Also, near as I can tell, Many Worlds seems to say the wave function represents other real universes. Not clear on that because the wave function is used before the measurement, and the Many Worlds seem to occur after. Also, the wave function interferes with itself, and the Many Worlds seem totally without interactions.

"Just possibilities?" Yes, that's all the wave function "represents". As it happens, I do think there's more to be understood behind the Schrodinger Equation. I'm just using the word "represents" in the more restricted sense in which we say "Let this mathematical expression represent [something real]". I am not implying the expression is reality, or that there can be nothing beyond the bare math.

Again, what's the point of trying to pigeon-hole me as some straw man? Are you so upset simply because I "defended Lubos" by pointing out he was right about once sentence? I assure you that doesn't put me "in his camp", nor mean that I agree with everything he writes -- because, frankly, I've never read anything he's written. (That said, I obviously don't buy the Many Words.)

[Probabilities] "That's a pretty freakin' big difference." Not really so much. Remember that classical physics includes waves and interference. Einstein showed that electromagnetic waves have particle-like properties. De Broglie showed that particles must have wave-like properties. Experiment shows that EM waves can knock electrons out of metals, which we would not expect of classical waves. And that electrons can create interference patterns, which we would not expect of classical particles.

A lot of people seem gobsmacked by superposition in QM. We understand new things by drawing relations to what we already know. Only after that can we also draw distinctions. Most people are just not very familiar superposition of waves in everyday life. That's where Dirac (with a background in electrical engineering) had an early advantage. That part just wasn't "weird" to him.

Yes, but you and I are not the only ones here. A substantial part of my audience is non-technical.

> "shut up and calculate": No, I am not in that crowd. Amazing you could jump to that conclusion based on a single sentence.

I think that's a not-entirely-unfair characterization of your stated position, that "What is the nature of the underlying reality?" is a "possibly meaningless" philosophical question, particularly since you made that statement in the context of defending Lubos, and introduced it with, "Sorry", which is a colloquialism that prefaces a dismissive statement.

> Are you so upset simply because I "defended Lubos" by pointing out he was right about once sentence?

No, I'm not upset. You and Lubos are simply both wrong. There are salient differences between a "collection of numbers" and a function.

> I simply see no reason the equations of Quantum Mechanics should answer such philosophical questions.

Those equations appear to be a supremely accurate description of reality. Do you really think it's unreasonable to suspect that they might contain some clues about the nature of the reality that they describe?

> [Probabilities] "That's a pretty freakin' big difference." Not really so much. Remember that classical physics includes waves and interference.

No, I don't think so. I think people are "gobsmacked" by the *combination* of interference and particle-like behavior. Either one by itself is not so mysterious. It's the combination that's weird. (That and entanglement.)

As a non-sequitur in this extensive comment thread, I just wanted to add my own layman perspective to the topic of the original post. It seems that each proposed interpretation of QM has aspects of reality and philosophy that it explains well, and other aspects that are more sketchy (esp. compared to alternative interpretations). I've long been a fan of MWI myself. It feels intuitive to me, and clarifies many of the so-called "paradoxes" that are common in QM descriptions to lay people.

All that said, the least satisfying aspect of MWI is the addition of the Born rule. For myself, for the moment, I'm content to let this aspect of QM remain as "not fully understood". And I love that people like Deutsch and Wallace are attempting to place the Born rule on firmer theoretical ground, in the context of MWI. That's an excellent direction to pursue, and I would predict that decades from now there will be some kind of well-agreed understanding throughout the field of the origin of the Born rule.

But I just want to put it out there, that the title of Ron's post ("The Trouble with Many Worlds") is not necessarily fatal for MWI, even if Deutsch and Wallace are wrong about the foundations of the Born rule.

Even without that (I would claim), MWI still remains a very strong contender for an understanding of quantum foundations. (Certainly much better than the original popular Copenhagen "conscious observation causes collapse", for example.)

I now return you to the comment thread technical debate on the merits of the Deutsch/Wallace proposed explanation of the Born rule...

I agree with you that Deutsch and Wallace's investigations are worthwhile, but I don't share your optimism that their efforts will ultimately succeed. Finding Born in MWI faces a fundamental problem: it depends entirely on branch weights (there is nothing else) but branch weights are inherently unknowable. Like I said above, it's the exact same problem that Bohmian positions have. Both branch weights and Bohmian positions are IPUs.

That said, I don't think that the MWI is entirely without merit. I think it's actually the correct interpretation from a "God's-eye view". But I, being a mortal, want to understand things from a mortal's-eye view, and for that I find MWI epistemologically, ontologically, and pedagogically inadequate.

BTW, my invocation of gods and mortals is not just a literary device. It's actually quite remarkable to me the extent to which these discussions resemble religious ones. The structural similarities of my discussions with Elliot and Alan in particular, and my discussions with Jimmy Weiss (my YEC friend) are really quite striking. I also learned recently that there is at least one bona fide "quantum cult" called "Ramtha's School of Enlightenment". (Google them. I don't want to improve their page rank by posting a link.) Their schtick is that because observation causes certain observables to actually become real, that you can learn to use your consciousness to *control* this process, and so do tricks like produce gold rings out of thin air with your mind.

Heh, TIL that RSE was behind the 2004 film "What the bleep do we know?"

I may have misled you! I'm agnostic about the specific effort of Deutsch/Wallace. (I'm not informed enough to have a useful opinion on their effort.) I was only expressing confidence that there will eventually be some satisfying explanation of the Born rule -- by somebody. And that the Deutsch/Wallace effort is the kind of approach (e.g., attempt to use decision theory) that I would expect from an effort that might eventually succeed (and convince everybody).

"Finding Born in MWI faces a fundamental problem: it depends entirely on branch weights"

That's just the specific Deutsch/Wallace approach. One could conceive that somebody else might invent some entirely different approach in the future to justify the Born rule within MWI.

@Don:the wave *function* is a function that predicts the evolution of the State Vector with time.

No, that's the Hamiltonian. The wave function is the state vector--more precisely, it's what you get when you pick a particular representation for the state vector in the space of square integrable functions from the configuration space to the complex numbers.

@Ron: The state vector is a function whose range is complex numbers and whose domain is the configuration space of a system.

Strictly speaking, that's the wave function. The state vector is just an abstract vector in a Hilbert space, with no commitment to any interpretation in terms of a particular configuration space or set of square integrable functions on it. For some Hilbert spaces, such as the Hilbert space of a qubit, there isn't even a representation in terms of square integrable functions, but you can still use the obvious representation in terms of column vectors.

The wave function simply adds time as another dimension of the domain, so these two terms are often safely interchangeable in many contexts.

The wave function can change with time, but it doesn't add time to the configuration space. Strictly speaking, the wave function "changing with time" really means there is a functional (which we could call a "wave function" if we were willing to muddy our terminology) mapping values of the parameter t, for "time", to wave functions, i.e., square integrable functions from the configuration space to the complex numbers.

Griffiths, for example, does not use the term "state vector" at all. It does not appear in the index of "Introduction to quantum mechanics (second edition)." (Why do I suddenly feel like Hermione Granger?) So at worst I think it's fair to say that the meaning of the term "state vector" is open to some interpretation.

Now, Griffiths does, of course, talk about vectors, mainly in Chapter 3 where he presents the QM formalism in terms of the linear algebra of Hilbert spaces. And in that context what you say above is certainly correct. However, on that view, the term "state vector" and "wave function" are *synonyms*. They are just two different ways of looking at the same underlying mathematical object.

But if you go back an look at the statement in which Dr. Guy introduced the term "state vector" into the conversation, he clearly intended "state vector" and "wave function" to be distinct:

> OK, so it's the *State Vector* that's just a (complex) number, and the wave *function* is a function that predicts the evolution of the State Vector with time.

I was trying to read this as charitably as I could. In particular, whatever Dr. Guy meant by "state vector" it had to be something that 1) evolved over time and 2) was predicted by the wave function. The only thing in my repertoire of quantum-mechanical concepts that fit these constraints was a mathematical description of a quantum system at a point in time, so that's what I assumed he meant. That interpretation is consistent with common usage, where one often sees people speak, for example, of a system being in a state |A> at some point in time, and then evolving over time into a different state |B> at some later point in time.

This is a big part of the pedagogical problem in QM. People throw these terms around in a very undisciplined way, while at the same time being haughty and snarky about it. They say things that are simply flat-out *wrong* (like "the state vector is (just) a complex number") and then when someone calls them on it they blame the caller-outer for not being smart enough to be able to understand that *of course* they didn't actually mean to say what they said (because what they said, taken at face value, is just *stupid*) they meant something *else*, something that is True and Profound, and which would be *obvious* if only their correspondent weren't such a complete *moron*.

Well, I say foo on that. If you're trying to explain this stuff to someone who is not an expert (which I think is a fair presumption of anything written in a *blog*) then the burden is on you to make yourself clear, and not on your audience to figure out what you actually meant to say.

BTW, being clear in this way is surprisingly challenging, and I don't blame anyone for failing. I fail all the time. The difference between me and Dr. Guy is that when I fail, I don't try to blame my audience.

I didn't say it did. I said it added time as another dimension of the *domain* (of the function).

> Strictly speaking, the wave function "changing with time" really means there is a functional...

Sure, but why make it more complicated than it needs to be? The wave function is a function of configuration and time, i.e. it is a function whose input is a whole bunch of (real) numbers, one of which is time, and whose output is a single complex number (if you ignore spin). Conceptually, that's all there is to it.

As a practical matter, the wave function is a horrifically complicated beast, and so to make the math tractable we make all kinds of simplifying assumptions about it, slice-and-dice it in various ways, and one of those ways is to look at it at a single point in time, and sometimes at a couple of discrete points or regions of space. But in the context of a discussion of MWI, the referent of the phrase "the wave function" is the wave function of the multiverse. That is a static entity. It never changes. (This is yet another reason that "the wave function" can only be seen from a "God's-eye view".)

@Ron: I think it's fair to say that the meaning of the term "state vector" is open to some interpretation.

If a particular textbook doesn't use it at all, that is not evidence that its meaning, as it is used in places that do use it, is open to interpretation.

The mathematical framework of vectors in a Hilbert space is standard QM, and I've never seen the term "state vector" used in any other way, if it's used at all. And the key point I was making is that such usage makes no commitment to any particular representation. It's similar to the way abstract tensor notation in General Relativity makes no commitment to any particular choice of coordinates. That lets important relationships be expressed in a way that is independent of representation (or coordinates in GR), which is a very nice property to have, particularly if, as many physicists do (and as I do), one takes the view that any physically meaningful quantity *must* be expressible in such a way.

on that view, the term "state vector" and "wave function" are *synonyms*

I don't think they are in general, because "wave function" requires a particular choice of representation, while "state vector" does not. See above. (Would you say, in the context of GR, that the terms "tensor" and "matrix" are synonyms? That would be analogous to your claim in what I just quoted.)

I don't have a copy of Griffiths so I can't look at the specifics of his presentation of vectors and Hilbert spaces in Chapter 3. If he uses the term "vector" to refer to particular column vectors in a particular choice of basis (for example, the spin-z basis of a qubit), it is true that that usage does pick out a particular representation, so it's not the same as the usage of "state vector" that I described above. (The abstract, representation-independent usage is usually associated with Dirac bra-ket notation, not column vector notation.)

if you go back an look at the statement in which Dr. Guy introduced the term "state vector" into the conversation, he clearly intended "state vector" and "wave function" to be distinct

Part of my purpose in posting was to correct what I saw as errors in Dr. Guy's posts. However, if he did intend "state vector" and "wave function" to be distinct, he was correct in that respect. He was just wrong in the particular distinct meanings he attributed to those terms (and in a bunch of other things as well; I'm certainly not claiming that none of your criticisms of Dr. Guy are valid).

in the context of a discussion of MWI, the referent of the phrase "the wave function" is the wave function of the multiverse. That is a static entity. It never changes

On the usual view of "wave function" as I've seen it in presentations of the MWI, it does. The universal wave function undergoes unitary evolution according to the overall Hamiltonian of the universe. And that unitary evolution includes entanglements being created by interactions between subsystems, such as "measured systems", "measuring devices", and "brains of observers". Such entanglements are how the MWI explains what we observe; without them the MWI has no predictive power.

Technically I suppose there could be a "God's eye view" where you consider the entire 4-D history of the universal wave function with time (since unitary evolution is deterministic and reversible, this can be done), but I don't see the usefulness of such a view in discussions of the MWI, since, as noted above, it does not help at all in explaining what we observe.

> If a particular textbook doesn't use it at all, that is not evidence that its meaning, as it is used in places that do use it, is open to interpretation.

That's true, but Griffiths is not just any old textbook. It's widely considered to be the leading textbook. So while the omission of a definition for "state vector" there is not probative, it is indicative.

I did a Google search to try to find a definitive definition of "state vector" *anywhere* and I couldn't find one. Wikipedia doesn't have one. Here's the closest it comes to defining the term:

"Mathematically, a pure quantum state can be represented by a ray in a Hilbert space over the complex numbers. The ray is a set of nonzero vectors differing by just a complex scalar factor; any of them can be chosen as a state vector to represent the ray and thus the state. A unit vector is usually picked..."

So...

> The mathematical framework of vectors in a Hilbert space is standard QM, and I've never seen the term "state vector" used in any other way, if it's used at all. And the key point I was making is that such usage makes no commitment to any particular representation.

Then see the Wikipedia quote above. It is using the word "ray" to mean the representation-independent mathematical object, and "state vector" to mean some particular representation of that ray.

> Would you say, in the context of GR, that the terms "tensor" and "matrix" are synonyms?

No, and that's a very good analogy. But it is simply not the case that there is agreement on the analogous terms in QM. Sometimes people use "vector" to play the role of "tensor", sometimes of "matrix". Where "ray" is used it is generally unambiguously analogous to "tensor", but it's rarely used.

> Technically I suppose there could be a "God's eye view" where you consider the entire 4-D history of the universal wave function with time (since unitary evolution is deterministic and reversible, this can be done), but I don't see the usefulness of such a view in discussions of the MWI, since, as noted above, it does not help at all in explaining what we observe.

I would say that the MWI does not help explain what we (which is to say, we mortals) observe *at all*. We mortals cannot obtain a God's-eye view, and on the mortal's-eye view, the MWI consists almost entirely of IPUs. The only reason that the MWI has any credibility at all is because it purports to take the math seriously. Well, if you're going to take the math seriously, then you need to take the math seriously, and the math says that the solution to the Schrödinger equation is a function of configuration and time. Now, you can do a partial evaluation of that function with respect to a point in time (whatever that might mean in a relativistic universe) and get a *different* function, one which is no longer time-dependent. And humans often find this a useful thing to do. But there's nothing in the math to indicate that this should have any physical significance. The *whole point* of the MWI is that you *don't* have to introduce random hacks like the Born rule or partial evaluation with respect to some particular variable in order to understand what is going on. If you're going to remain true to that principle, I don't see any way you can see the wave function as anything other than a static 4-D entity.

@Ron: "The *whole point* of the MWI is that you *don't* have to introduce random hacks like the Born rule"

I'm not sure that I would have described the situation that way. The original MWI (as I understand it) does seem to have the Born rule as an additional, not well-understood, "hack". This is why the attempt by Deutsch/Wallace is so interesting, and would be so valuable (if it succeeded). Because the standard MWI does not yet have a well-justified explanation for the Born rule.

"I don't see any way you can see the wave function as anything other than a static 4-D entity."

This part seems to me to head into the general territory of why there is an arrow of time (and a "present moment") at all. That's a great question, but it's a much larger scoped question than MWI, or even than just QM interpretations.

> The original MWI (as I understand it) does seem to have the Born rule as an additional, not well-understood, "hack".

No. In the MWI there is no collapse. Everything is completely deterministic. There are no probabilities, so the Born rule is not only unnecessary, it's *incoherent*. (That's why Deutsch's argument is framed in terms of the deterministic parts of decision theory rather than probability theory.)

Of course, the Born rule is also an empirically observed *fact*, which is why the MWI is sometimes criticized as "failing to make contact with reality" or words to that effect (I don't have time to actually look up the quote right now).

@Ron: I don't think you're being fair to MWI. At some point, you need to predict the outcomes of experiments. E.g., If you run a million of these tests, your counter is going to report ("very close to") 3/4 "A" result and 1/4 "B" result. Every QM interpretation needs to yield the same predictions. MWI gets those predictions by a Born rule. MWI without a Born rule doesn't predict anything. It isn't even a candidate QM interpretation. So it's far from "unnecessary".

Whether it's "incoherent" is a somewhat different question. There is a somewhat open question of, if the Born rule gives probabilities, in MWI what exactly are they supposed to be probabilities of? That's not an easy question to answer. (And it begins to feel like it might lead back to "consciousness" and "observers" in some way -- which is the opposite of the goal here.)

But I have never seen anyone advocate for an MWI interpretation without a Born rule. If that's what you think, no wonder it didn't hold much appeal for you. I don't know what to point you to, but wikipedia, for example, has lots of discussion about MWI advocates attempting "to derive the Born rule, rather than just conventionally assume it". Whether derived or assumed, that's very different from your claim that "MWI has no Born rule" at all.

I'm going to respond out of order, because I've left you with a very wrong impression:

> If that's what you think, no wonder it didn't hold much appeal for you.

It's not that the MWI doesn't "hold much appeal." It's not that I dislike it. It's not even that I think it's *wrong*; I don't. It isn't. The SE really does imply that there are multiple worlds. But these multiple worlds also really are IPUs from a mortal's-eye point of view, and I happen to be a mortal. That's just the way it is, whether I or you or anyone else likes it or not.

> I have never seen anyone advocate for an MWI interpretation without a Born rule.

I'm not sure what you mean by "a Born rule". You say that is if there were more than one. The debate is not over whether *the* Born rule needs to be part of the story; obviously everyone agrees that it does. The disagreement is over whether the Born rule needs to be accepted as an axiom, or whether it can be derived from something else (specifically, from the SE, because that's all there is) in the same way that the structure of the multiverse itself can. Deutsch and Wallace say that it can. I say they've just snuck the Born rule in as an axiom in disguise, specifically, in the form of assuming branch indifference.

Furthermore, *if* you concede that you have to accept the Born rule as an axiom (in one form or another) and that you really can't deriving it from the SE, then I say that you've lost the moral high ground from which the MWI is typically defended. The story goes: you should accept this outrageous assertion that the universe is peeling apart trillions of times each second (at least!) because that's what the math says. And it's true, that *is* what the math says, or at least it's what the SE says. The problem is that the SE by itself doesn't explain all the data. IMHO of course. Deutsch and Wallace would disagree. Elliot and Alan clearly do. But so far neither of them has been able to explain to me why I should accept branch indifference as a pre-requisite for rationality. Until and unless someone can do that I don't see any reason to *prefer* MWI over the alternatives. And in particular I don't see why I should prefer it over QIT (which is really the same as MWI but looking at it from a mortal's-eye POV).

@Ron: "The disagreement is over whether the Born rule needs to be accepted as an axiom, or whether it can be derived from something else"

Sure, although I might word it as: everyone would love to derive the Born rule from something more fundamental. But only some people think they have such a derivation. The others, realizing that you can't do without it, reluctantly accept it as an axiom. (I don't think anyone wants the Born rule to be only an axiom.)

"then I say that you've lost the moral high ground from which the MWI is typically defended"

Well, that may be true. :-)

I think at the moment, I prefer to view it as: the MWI explanation is currently incomplete. 3/4 of the theory "seems right", but the Born rule isn't yet as well explained as it needs to eventually be. Hopefully some future work (perhaps Deutsch/Wallace, or maybe some different approach) will clean up this "loose end".

(Or else, perhaps it's instead like failure to find the ether, and the speed of light is constant, and to resolve this tiny problem in a corner of physics we need to completely reconceptualize all of space and time. You can never tell what the eventual resolution of a nagging theoretical problem might become.)

"And in particular I don't see why I should prefer it over QIT"

Yeah, I don't have a strong opinion about that. I certainly prefer MWI to "Copenhagen conscious observation collapse", which is just incoherent. And pilot waves don't seem (to me) to add any value (over MWI) either. But once you're in the space of actual scientific "explanations", I don't especially have any arguments to offer you for MWI vs. QIT.

So: yeah, this last comment doesn't bother me as much as your original post (title). :-)

> I certainly prefer MWI to "Copenhagen conscious observation collapse", which is just incoherent.

That's one of the beautiful things about QIT: it makes this coherent. QIT is the *only* interpretation that really takes the math seriously. In particular, it's the one interpretation that takes the no-cloning theorem seriously, and the consequent fact that in order to create a classical system from a quantum one you have to discard information. That is the reason that you have to choose between the God's-eye view and the mortal's-eye view. The God's-eye view is what you get when you don't discard any information, and the mortal's-eye view is what you get when you discard the information necessary to create a sub-system that contains copyable classical information. That in turn is a pre-requisite for consciousness.

So the truth really depends entirely on how you choose to look at it, and at what point you choose to discard the information needed to produce classicality. The only point at which you are *forced* to make that choice is when you reach your own consciousness. So in this respect, the Copenhagenists actually have a point!

This is one of the most frustrating things about this whole debate. It's like the blind men and the elephant. They're all right, and they're all wrong. The truth really does depend entirely on the point of view you choose to take.

@Ron:see the Wikipedia quote above. It is using the word "ray" to mean the representation-independent mathematical object, and "state vector" to mean some particular representation of that ray.

Yes, this is a valid point; strictly speaking, any vector on the same ray in a Hilbert space represents the same physical state, so we have to pick one in order to actually do any math. However, the sense in which the particular state vector we pick "represents" the physical state is *not* the same as the sense of "representation" I was using before. Picking a particular vector in a ray in Hilbert space to represent a physical state still does not pick out any particular basis or mathematical form for writing down the state vector; we can just write a ket (or a bra for a vector in the complex conjugate space), and that doesn't tell us whether, for example, we are using the position or the momentum representation for the spatial motion of a particle, or what spin axis (z or x or whatever) we are using to write down the spin part of the state, etc. When I was using the term "representation" before, it was to describe the latter type of choice: writing a wave function specifically in the position representation, or writing a qubit's state specifically in the spin-z basis, for example. That choice of representation is separate from (and comes after) the choice of which vector in a Hilbert space ray to use to describe the physical state.

I agree there isn't a single standard terminology for all this, but the two concepts I have described are definitely distinct and it is worth keeping track of the distinction somehow.

@Ron:if you're going to take the math seriously, then you need to take the math seriously, and the math says that the solution to the Schrödinger equation is a function of configuration and time.

More precisely, it says that the *time derivative* of the wave function is given by the Hamiltonian operating on that wave function. But time in this formalism is not a coordinate--it's not a label for points in spacetime. It's a parameter; the wave function is really a functional parameterized by time--it's a continuous set of functions on configuration space, each one mapped to a particular value of the parameter t. You can view this as "a function of configuration and time" in the sense that we have to be able to define a derivative of the wave function with respect to time (the parameter), but I don't think that the further inferences you are trying to draw from "a function of configuration and time" are justified. See below.

I don't see any way you can see the wave function as anything other than a static 4-D entity.

No, it isn't, because configuration space is not ordinary 3-space, and, as above, "time" is not a label for points in spacetime, it's a parameter. You could, I suppose, imagine an infinite "stack" of configuration spaces, each one labeled by a value of t, and view the wave function as a function on the entire stack, with inputs t (labeling which item in the stack) and x (labeling which configuration space point in that item). But this will not be a "static 4-D entity" except in the single special case where your entire quantum system is a single spinless particle moving in 3 dimensions. And viewing it as static ignores the fact that, as above, the Schrodinger equation relates the time derivative of psi to the Hamiltonian operating on psi, which doesn't seem like a "static" equation; it seems like a dynamic equation, telling us how psi changes with time.

> time in this formalism is not a coordinate--it's not a label for points in spacetime. It's a parameter

So? What difference does it make what the semantics of the time parameter are? It's still an input to the function.

> the wave function is really a functional parameterized by time

I dispute your use of the word "really" here. A functional is just a particular kind of function, and whether a function is a functional depends on how you choose to look at it. Any function of more than one argument can be seen as a functional.

For a single particle in a one-dimensional space (the classic "particle in a box") the wave function is a function of x and t. You can choose to look at it as a functional, where you plug in t and get out a new function that is a function only of x, or you can choose to look at it as a function of two arguments, or as a function of a single argument that is an ordered pair. All of these things are mathematically equivalent.

It just so happens that the parameter we call t corresponds to our experience of what we call "time" and the parameter we call x corresponds to our experience of what we call "space". You can choose to view t as a parameter, and the resulting function of x as telling you the "state" of the particle at a "point" in time. But you could just as well choose to view x as a parameter, and the resulting function telling you the evolution of the amplitude of the particle over time at a point in space. It all depends on the point of view you choose to take. And, of course, once you throw relativity into the mix then it *really* depends on the point of view you choose to take because space and time are literally interchangeable.

> it seems like a dynamic equation, telling us how psi changes with time.

Only from a particular point of view, i.e. a mortal's-eye POV bound to a specific inertial reference frame. From a God's-eye point of view, which is the POV you have to take in order to see the multiverse, it's static.

@Ron: You can choose to view t as a parameter, and the resulting function of x as telling you the "state" of the particle at a "point" in time. But you could just as well choose to view x as a parameter, and the resulting function telling you the evolution of the amplitude of the particle over time at a point in space.

But the first view, viewing t as a parameter labeling points in time, works for any quantum system, while the second view, viewing x as a parameter labeling points in space, only works for the special case of a single spinless particle. Time remains time (corresponding to our experience of time, as you say) for any quantum system, but configuration space is not ordinary space for any quantum system other than a single spinless particle.

once you throw relativity into the mix then it *really* depends on the point of view you choose to take because space and time are literally interchangeable.

If we're going to talk about relativistic QM, much of what we've said so far is irrelevant. Relativistic QM, i.e., quantum field theory, does not use the Schrodinger equation and does not have wave functions (more precisely, it doesn't have them as fundamental entities in the formulation of the theory, only as constructions that can be used in particular cases if you want to use a non-relativistic approximation). I would have no problem switching discussion to QFT, but I note that in the vast literature on interpretations of QM, I see little or no discussion of QFT, and virtually all discussion is based on non-relativistic QM (which to me is a huge defect of all that literature).

Also, I don't think it's right to say "space and time are literally interchangeable" in relativity. Spacelike and timelike vectors in relativity are physically and mathematically distinct; you can't transform one into the other. (And there is a third category, null vectors, which is also distinct from the other two.) Talk about "space and time getting mixed together" by Lorentz transformations is vague and often misleading and IMO should be avoided.

@Ron:From a God's-eye point of view, which is the POV you have to take in order to see the multiverse, it's static.

I disagree. A "God's eye point of view" can be taken simply by viewing the wave function--by which I mean the wave function at a particular value of the time parameter t--as describing the entire universe. Then the evolution of the wave function as t changes is just the time evolution of the entire universe. If you pick out different entangled terms in the universal wave function as describing different "worlds", you have a multiverse of worlds all evolving in time.

Note that I am not saying you *cannot* take a "static" point of view in which you view the entire time evolution as one static thing. I am only saying you are not *required* to take that point of view in order to see a multiverse. If the MWI is correct, there is a multiverse at every instant of time t (meaning every value of the parameter t).

A "mortal's eye view" means restricting attention to a particular subsystem of the universe that one is interested in (usually the subsystem that one is doing experiments on) and ignoring the rest. That prevents you from seeing a multiverse simply because you are restricting to a particular subsystem (meaning one particular entangled term in the universal wave function).

> configuration space is not ordinary space for any quantum system other than a single spinless particle

That doesn't matter. From a mathematical point of view, psi is still a function of a bunch of numbers, and you can partially evaluate that function with respect to any subset of the input variables, at least in principle. The special nature of time is a reflection of the *physics* (and specifically a reflection of how *we experience* the physics), not the *math*. This matters if you're going to hang your interpretational hat on the premise that you're taking the math seriously.

> If we're going to talk about relativistic QM, much of what we've said so far is irrelevant.

You understand QFT better than I do. But I am given to understand that the Dirac equation is structurally the same as the SE, just with some higher order terms in the Hamiltonian. If that's the case, then I don't see how anything that has been said to this point would not apply.

> you are not *required* to take that point of view in order to see a multiverse

Fair enough. Let me try making this point a different way: a DVD is a static object. You can choose to put that static object into a DVD player and watch the movie as a dynamic thing, but that does not change the fact that the DVD is static, even while the movie is playing. With modern technology you can even play "the same" DVD from multiple temporal points of view simultaneously. What matters is that the DVD contains information that does not change over time. You can extract and render that information in different ways, some of which play out over time, but none of those processes add information to the DVD, nor can they change the information on the DVD.

The wave function is analogous to the DVD, with the additional proviso that from a single frame you can reconstruct the entire DVD (assuming QM is correct).

So if you are God, you can choose to watch "The Multiverse: the Movie" as a process playing out over time, or you can choose to take the DVD out of the player and see it as a static object. You can also reconstruct all the information on the cosmic DVD from a single time slice. But (again assuming physics is unitary) nothing can change the content of the cosmic DVD, not even God (assuming God is constrained by the laws of physics). Hence, the wave function, viewed in terms of its information content (and that is all the wave function is -- a repository of information), is static.

> A "mortal's eye view" means restricting attention to a particular subsystem of the universe that one is interested in (usually the subsystem that one is doing experiments on) and ignoring the rest. That prevents you from seeing a multiverse simply because you are restricting to a particular subsystem (meaning one particular entangled term in the universal wave function).

Yes, exactly. But if you are not God, if you are classical, if you are a process that relies on copying information (and you are), then you have no choice but to take the mortal's-eye point of view. You can become aware in the abstract that the God's-eye point of view exists, but you cannot take it yourself. (Actually, if God is constrained by the laws of physics then not even God can take a God's-eye point of view because that would violate the non-cloning theorem. The only entity that can take a God's-eye point of view is the wave function itself.)

I enjoyed (and appreciated) the analogy. Although of course, the old idea of a "block universe" (as one perspective on the "illusion" of time) doesn't even depend on quantum mechanics. That's a philosophical concept even in a Newtonian universe.

"you have no choice but to take the mortal's-eye point of view"

I suspect nobody disagrees that conscious experience only observes sensory data about one "world" (in MWI). I'm not sure about that "taking a point of view" thing, though. E.g., the book Flatland was an attempt to analogize explaining our 3D experience to hypothetical 2D creatures, in order to "take the point of view" of explaining four spacial dimensions to ourselves. I thought the whole point of "taking the point of view" was to use analogy and imagination to put yourself in the place of someone else, that you in actual fact are not.

Perhaps I've lost the thread. If I grant you the ("point of view of the") DVD, and I grant you that we only experience the thread of a single world ... what implication or conclusion are you trying to make about the MWI QM interpretation?

@Ron: I am given to understand that the Dirac equation is structurally the same as the SE, just with some higher order terms in the Hamiltonian.

This is basically true of the Dirac equation as it was originally constructed, to be an analogue of the Schrodinger Equation for a spin-1/2 particle. However, the Dirac equation in this interpretation is not QFT. It's an early attempt to construct a quantum theory of a relativistic particle with spin 1/2, which didn't work out.

The fundamental difference between QFT and non-relativistic QM is that the basic ontology changes. Quantum fields are not wave functions. They're *operators*. Basically a quantum field theory is an assignment of a set of operators to each point in spacetime, subject to some equations that the operators have to obey (such as that the operators assigned to spacelike separated events have to commute).

You can reinterpret the Dirac equation as an equation that has to be satisfied by quantum field operators (for example, the operators for the electron field in quantum electrodynamics), but if you do that it's no longer an analogue of the Schrodinger Equation. (This process of reinterpretation, which has to be done with all field equations in QFT if you're using the canonical quantization procedure, is what is often referred to as "second quantization" in textbooks.)

As far as how all this ties in with the various different interpretations of QM, as I said, I have seen virtually no discussion of that at all in the literature. So I don't know that there is any generally accepted way to even frame the MWI in the context of QFT.

Flatland is not a bad analogy, but it's flawed in one very significant way. Going from two dimensions to three is very similar to the process of going from three to four. That is not true for going to the classical to the quantum. There is a fundamental unbridgeable chasm in the form of the no-cloning theorem. The only way you can take any kind of point of view about anything is to somehow copy information about that thing into your brain. But quantum information can't be copied.

The no-cloning theorem is a much more profound result than seems to be widely appreciated. It's on a par with Bell's theorem IMHO, but Bell get's all the attention. The no-cloning theorem rules out our ability to take a God's-eye view in the same way that Bell's theorem rules out local hidden variables. It insures that no experiment we can ever do will demonstrate the existence of alternate universes.

Here's a spiffy (IMHO) illustration of this: Deutsch and Wallace's derivation of the Born rule depends crucially on this idea of branch weight. In other words, it depends on the idea that if you do a quantum experiment at odds other than 50-50, that the "weight" of the branch you end up on has to *matter* in some way, in particular, that copies/versions of you that end up in "heavier" branches "count" more than those who end up in lightweight branches for the purposes of rational decision-making.

A natural question to ask, then, is: can branch weights be measured? In particular, can we measure our own branch weight? The answer turns out to be no, you can't. Branch weights are an IPU -- provably, necessarily -- just like parallel universes themselves. I'll leave the proof as an exercise. It's really simple. Just ask yourself: suppose I could measure my own branch weight, how could I use that information to make a copy of an unknown quantum state? Hint: think in terms of polarization.

@Ron:A natural question to ask, then, is: can branch weights be measured? In particular, can we measure our own branch weight? The answer turns out to be no, you can't. Branch weights are an IPU -- provably, necessarily -- just like parallel universes themselves.

Since I have been disagreeing with some things you've said, I should emphasize that I am not disputing any of this; I agree with it, and I agree that it is a valid criticism of the MWI.